ieee cis outstanding ph.d. dissertation award...

IEEE CIS Outstanding Ph.D. Dissertation Award Nomination

Nominee Dr. Zhi-Hui Zhan South China University of Technology [email protected]

Nominator Prof. Qingfu Zhang City University of Hong Kong [email protected]

Referee Prof. Derong Liu University of Illinois at Chicago [email protected]

Prof. C. A. C. Coello CINVESTAV-IPN, Mexico [email protected]

Prof. Jun Wang City University of Hong Kong [email protected]

Prof. C. L. Philip Chen University of Macau [email protected]

Prof. Chin-Teng (CT) Lin National Chiao-Tung University [email protected]

Contents

Part I: Nomination Letter

Part II: Referee Letters

Part III: Ph. D. Dissertation

Part I: Nomination Letter

Nomination Letter Nominator: name, affiliation and email address of nominator

Name: Qingfu Zhang

Affiliation: City University of Hong Kong

Email Address: [email protected]

Nominee: name, affiliation, postal address and email address of

nominee

Name: Zhi-Hui Zhan

Affiliation: South China University of Technology

Postal Address: Room 515, School of Computer Science and

Engineering, South China University of

Technology, Da-Xue-Cheng, Guangzhou,

Guangdong, P. R. China, 510006

Email Address: [email protected]; [email protected]

Dissertation: title of the dissertation, institution in which the

degree was conferred

Title: Research into Machine Learning Aided Particle

Swarm Optimization and Its Engineering

Application

Institution: Sun Yat-sen University

Proposed Citation: provide suggestion for the complete, correct

and succinct citation. The Awards Committee reserves the right

to make any necessary change on the citation.

2017: Zhi-Hui Zhan, “Research into Machine Learning Aided Particle

Swarm Optimization and Its Engineering Application,” Sun Yat-sen

University, China, 2013.

List of Publication in journals and conference proceedings

generated by the research reported in the PhD dissertation;

13 most related papers include 9 IEEE Transactions papers.

3 of them are ESI Highly Cited Papers

Publications Related to Chapter 2

[1]. Zhi-Hui Zhan, J. Zhang, Y. Li, and H. Chung, “Adaptive particle swarm optimization,” IEEE Transactions on Systems, Man, and Cybernetics--Part B, vol. 39, no. 6, pp. 1362-1381, Dec. 2009. [Related to Chapter 2: Propose the adaptive PSO]

ESI Highly Cited Paper

Google Scholar Citation 1026 times, SCI Citation 511 times

The Top 2 cited paper of this journal in recent 10 years, since 2006

[2]. Y. L. Li, Zhi-Hui Zhan (Corresponding Author), Y. J. Gong, W. N. Chen, J. Zhang, and Y. Li, “Differential evolution with an evolution path: A DEEP evolutionary algorithm,” IEEE Transactions on Cybernetics, vol. 45, no. 9, pp. 1798-1810, Sept. 2015. [Related to Chapter 2: Extend the adaptive idea to DE]



[3]. Zhi-Hui Zhan, J. Zhang, Y. Li, and Y. H. Shi, “Orthogonal learning particle swarm optimization,” IEEE Transactions on Evolutionary Computation, vol. 15, no. 6, pp. 832-847, Dec. 2011. [Related to Chapter 3: Propose the orthogonal learning PSO]

ESI Hot Paper, ESI Highly Cited Paper



[4]. Y. H. Li, Zhi-Hui Zhan (Corresponding Author), S. Lin, J. Zhang, and X. N. Luo, “Competitive and cooperative particle swarm optimization with information sharing mechanism for global optimization problems,” Information Sciences, vol. 293, no. 1, pp. 370-382, 2015. [Related to Chapter 3: Extend the orthogonal learning strategy to competitive and cooperative strategy]




[5]. Zhi-Hui Zhan, J. Li, J. Cao, J. Zhang, H. Chung, and Y. H. Shi, “Multiple populations for multiple objectives: A coevolutionary technique for solving multiobjective optimization problems,” IEEE Transactions on Cybernetics, vol. 43, no. 2, pp. 445-463, April. 2013. [Related to Chapter 4: : Propose the co-evolutionary multiswarm PSO for MOP]



[6]. Y. L. Li, Y. R. Zhou, Zhi-Hui Zhan (Corresponding Author), and J. Zhang, “A primary theoretical study on decomposition-based multiobjective evolutionary algorithms,” IEEE Transactions on Evolutionary Computation, vol. 20, no. 4, pp. 563-576, Aug. 2016. [Related to Chapter 4: Theoretical study on multiobjective evolutionary algorithms]

[7]. H. H. Li, Z. G. Chen, Zhi-Hui Zhan (Corresponding Author), K. J. Du, and J. Zhang, “Renumber coevolutionary multiswarm particle swarm optimization for multi-objective workflow scheduling on cloud computing environment,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2015), Madrid, Spain, Jul. 2015, pp. 1419-1420. [Related to Chapter 4: Apply the CMPSO algorithm to cloud computing resources scheduling]


[8]. Zhi-Hui Zhan and J. Zhang, “Orthogonal learning particle swarm optimization for power electronic circuit optimization with free search range,” in Proc. IEEE Congr. Evol. Comput. (CEC 2011), New Orleans, Jun. 2011, pp. 2563-2570. [Related to Chapter 5: Propose to use OLPSO to solve PEC]

[9]. M. Shen, Zhi-Hui Zhan (Corresponding Author), W. N. Chen, Y. J. Gong, J. Zhang, and Y. Li, “Bi-velocity discrete particle swarm optimization and its application to multicast routing problem in communication networks,” IEEE Transactions on Industrial Electronics, vol. 61, no. 12, pp. 7141-7151, Dec. 2014. [Related to Chapter 5: Engineering application of PSO]


[10]. X. F. Liu, Zhi-Hui Zhan (Corresponding Author), D. Deng, Y. Li, T. L. Gu, and J. Zhang, “An energy efficient ant colony system for virtual machine placement in cloud computing,” IEEE Transactions on Evolutionary Computation, DOI: 10.1109/TEVC.2016.2623803. 2016. [Related to Chapter 5: Engineering application]

[11]. Y. L. Li, Zhi-Hui Zhan (Corresponding Author), Y. J. Gong, J. Zhang, Y. Li, and Q. Li, “Fast micro-differential evolution for topological active net optimization,” IEEE Transactions on Cybernetics, vol. 46, no. 6, pp. 1411-1423, Jun. 2016. [Related to Chapter 5: Engineering application]

Publications Related to Chapter 1&6

[12]. J. Zhang (Supervisor), Zhi-Hui Zhan, Y. Lin, N. Chen, Y. J. Gong, J. H. Zhong, H. S. H. Chung, Y. Li, and Y. H. Shi, “Evolutionary computation meets machine learning: A survey,” IEEE Computational Intelligence Magazine, vol. 6, no. 4, pp. 68-75, Nov. 2011. [Related to Chapters 1&6: Survey of EC&ML]


[13]. Zhi-Hui Zhan, X. Liu, H. Zhang, Z. Yu, J. Weng, Y. Li, T. Gu, and J. Zhang, “Cloudde: A heterogeneous differential evolution algorithm and its distributed cloud version,” IEEE Transactions on Parallel and Distributed Systems, vol. 28, no. 3, pp. 704-716, March. 2017. [Related to Chapter 6: Future work]

Other information, if applicable, that can be used for the

evaluation of the PhD dissertation.

Abstract of the Ph. D dissertation Particle swarm optimization (PSO) is a kind of simple yet powerful optimization technique. When compared with other evolutionary computation (EC) algorithms such as genetic algorithm (GA), PSO is simper in the algorithm structure, easier in the implementation, and faster in convergence. Therefore, PSO has good application prospect in various science and engineering optimization problems, attracting great interesting and attention from researchers all over the world. During the almost two decades’ development since PSO was invented in 1995, some key problems as follows emerge and call for urgent solutions.

1) The PSO performance strongly relies on the parameter and operator in different evolutionary states. How to be aware of the evolutionary states and adaptively control the parameter and operator to obtain better algorithm performance is a hot yet difficult research topic in PSO community.

2) Although PSO can obtain a reasonable solution fast for various problems, the fast convergence speed makes PSO easy to be trapped into local optima, especially in complex multimodal optimization problem. How to develop a PSO variant with both faster convergence speed and strong global search ability is a significant yet challenging research topic in PSO community.

3) When applying PSO to applications such as multi-objective optimization and engineering optimization problem in practices, how to utilize and combine the problem characteristics so as to efficiently solve the practical problem is still a challenging problem in extending PSO to real-world applications.

In response to these issues, this dissertation carries out innovative researches into the PSO parameter adaptation control, operator orthogonal design, and population co-evolutionary interaction. In order to make these researches efficient, this dissertation points out that the characteristics of population-based search and iteration-based evolution in PSO provide a mass of search data and historical data during the evolutionary process. As machine learning (ML) technique is a powerful tool for obtaining useful information from large amounts of data, using ML technique to analyze, process, and utilize these data has great significance to aid PSO algorithm design and so as to improve the algorithm performance. In view of this, this dissertation conducts researches into ML aided PSO and its engineering application. The main works are to apply the techniques and ideas such as statistical analysis, orthogonal design and prediction, and ensemble learning in the ML field to aid PSO design, improving the algorithm performance, and extending its applications.

The main innovative contributions of this thesis are as follows: (1) Propose a statistical analysis based adaptive PSO (APSO) to make the algorithm can act properly according to different states, enhancing the algorithm versatility.

The parameter and operator requirements for PSO are different in different evolutionary states. By using the strong ability of ML technique in obtaining useful information from mass data, this dissertation proposes to make statistical analyses on the population distribution data and fitness data of PSO during the evolutionary process. This results in a novel evolutionary state estimation (ESE) method that can classify different evolutionary states efficiently. By using the ML technique aided ESE method, APSO can adaptively control the parameters and operators according to different requirements in different states, improving the PSO performance and enhancing the PSO versatility in different search environments. (2) Propose an orthogonal design and prediction based orthogonal learning PSO (OLPSO) to enhance the algorithm global search ability in complex optimization.

As the learning strategy in traditional PSO can not sufficiently utilize the information in the personal experience and the neighborhood experience, this dissertation proposes a novel orthogonal learning (OL) strategy for the particle to construct a promising exemplar to guide the flying. The OL strategy is based on the orthogonal experimental design technique in ML that can efficiently discover useful information in the personal and neighborhood experiences and predict promising combination of these two experiences. Therefore, OLPSO can obtain both fast convergence speed and strong global search ability. The promising performance of OLPSO makes it an efficient tool for complex and multimodal optimization problems. (3) Propose a co-evolutionary multi-swarm PSO (CMPSO) inspired by the ensemble learning idea in ML, enhancing the performance in multi-objective optimization.

The ensemble learning method in ML is to use multiple classifiers to enhance the classification ability. Inspired by such multiple learners’ idea, this dissertation designs a novel optimization framework as multiple populations for multiple objectives (MPMO) when using EC algorithms to solve multi-objective optimization problems (MOP). Based on the MPMO framework, the CMPSO algorithm on the one hand avoids the fitness assignment problem which caused by considering all the objectives together, and on the other hand searches sufficiently in different areas of the Pareto front (PF) by the guidance of each objective. Moreover, CMPSO uses a novel external shared archive for the communication and co-evolution of different swarms, so as to make the non-dominated solutions cover along the whole PF efficiently, enhancing the performance in MOP. (4) Apply OLPSO to the power electronic circuits (PEC) design problem, extending the engineering application fields of PSO.

The PEC design problem is a complex engineering application problem for that it involves lots of components such as resistors, capacitors, and inductors, which are all needed optimally designed so as to obtain good circuit performance. This dissertation on the one hand extends the traditional PEC optimization model by introducing free search range for the components. Although this new model makes PEC much closer to real-world application, it brings great challenges to current optimization methods. Therefore, this dissertation on the other hand proposes to apply the powerful ML aided OLPSO to optimize PEC with free search range. The successes of OLPSO in

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The Short CV of the Nominee Zhi-Hui Zhan (March 2017)

Zhi-Hui Zhan Professor, Ph.D. Research Interests

Computation Intelligence Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), Genetic Algorithm

(GA), Differential Evolution (DE), Brain Storm Optimization (BSO) Cloud Computing and Big Data

Large Scale Resources Scheduling, Multiobjective Optimization, Dynamic Optimization Intelligent Application

Wireless Sensor Network, Scheduling and Control, Intelligent Systems

Work and Education 01/2016 – Now Professor, School of Computer Sci. and Eng. South China Univ. Technology 01/2015 – 12/2015 Associate Professor, School of Advanced Computing Sun Yat-sen University 07/2013 – 12/2014 Lecturer, School of Information Sc. and Technology Sun Yat-sen University 09/2009 – 06/2013 Ph. D., Computer Application Technology Sun Yat-sen University 09/2003 – 07/2007 Bachelor of Science, Computer Sci. and Techno. Sun Yat-sen University

Awards and Honors

2016, Pearl River Scholar Young Professor 2016, Elsevier Most Cited Chinese Researchers in Computer Science 2015, Elsevier Most Cited Chinese Researchers in Computer Science 2014, Elsevier Most Cited Chinese Researchers in Computer Science 2014, Natural Science Found for Distinguished Young Scholars of Guangdong Province, China 2015, Pearl River New Star in Science and Technology 2014, Guangdong Province Outstanding Dissertation Award 2013, China Computer Federation (CCF) Outstanding Dissertation Award

Projects

2015-2017, National Natural Science Foundations of China (NSFC) PI, 270,000RMB 2014-2018, NSF of Guangdong Province for Distinguished Young Scholars PI, 1,000,000RMB 2015-2017, Project for Pearl River New Star in Science and Technology PI, 300,000RMB 2015-2016, Fundamental Research Funds for the Central Universities PI, 300,000RMB 2013-2015, National High-Tech. Research and Develop. Program (863) of China Co-PI, 26,800,000RMB

Academic Activities and Services

The Committee of Machine Learning in CAAI Committee Member The Committee of Artificial Intelligence and Pattern Recognition in CCF Committee Member The 7th Int. Conf. on Information Science and Technology (ICIST 2017) Special Sessions Co-Chair IEEE World Congr. Comput. Intell. (WCCI 2014), Special Session Proposer Int. Conf. Machine Learning & Cybern. (ICMLC 2013) Invited Session Organizer Program Committee Member of 10+ International Conferences, Including:

The Thirty-First AAAI Conference on Artificial Intelligence (AAAI 2017 ) Evolutionary Machine Learning (EML) track of The ACM GECCO 2017 International Conference on Swarm Intelligence (ICSI 2016/2015/2014) Asia-Pacific Services Computing Conference (APSCC 2014) Conf. Technologies and Appl. of Artificial Intell. (TAAI 2015/2014/2013) Int. Conf. Software, Multimedia and Communication Engineering (SMCE 2015)

Regular Reviewer of 20+ International Journals, Including: Since 2009 IEEE Transactions on Evolutionary Computation Since 2012 IEEE Transactions on Cybernetics Since 2009 IEEE Transactions on Industrial Electronics Since 2012 IEEE Transactions on Computational Intelligence and AI in Games Since 2012 IEEE Computational Intelligence Magazine Since 2011 Information Sciences

Invited Talks

01/2017, Int. Workshop on Intell. Opt. and Social Computing (IWIOSC 2017), Changsha, China 10/2016, The 11th Int. Conf. on Bio-inspired Computing: Theor.&Appl. (BIC-TA 2016), Xian, China 08/2014, The First Young Scholar Forum of CAAI, Nanchang, China 06/2015, The 2nd Evolutionary Computation and Learning Forum (ECOLE 2015), Nanjing, China

Monograph

[1]. J. Zhang, Zhi-Hui Zhan, W. N. Chen, J. H. Zhong, N. Chen, Y. J. Gong, R. T. Xu, and Z. Guan, Computation Intelligence, Tsinghua University Press, November, 2011. (Chinese)

[2]. J. Zhang, W. N. Chen, X. M. Hu, Y. Lin, W. L. Zhong, Zhi-Hui Zhan, and T. Huang, Numerical Computing，Tsinghua University Press, July, 2008. (Chinese)

ESI Hot Paper [1]. Zhi-Hui Zhan, J. Zhang, Y. Li, and Y. H. Shi, “Orthogonal learning particle swarm

optimization,” IEEE Transactions on Evolutionary Computation, vol. 15, no. 6, pp. 832-847, Dec. 2011. (IF=5.908; Citation: Google Scholar 351 times, SCI 196 times; The Top 3 cited paper of this journal in recent 5 years, since 2011)

ESI Highly Cited Paper [2]. Zhi-Hui Zhan, J. Zhang, Y. Li, and H. Chung, “Adaptive particle swarm optimization,”

IEEE Transactions on Systems, Man, and Cybernetics--Part B, vol. 39, no. 6, pp. 1362-1381, Dec. 2009. (IF=4.943; Citation: Google Scholar 1026 times, SCI 511 times; The Top 2 cited paper of this journal in recent 10 years, since 2006)

[3]. Zhi-Hui Zhan, J. Li, J. Cao, J. Zhang, H. Chung, and Y. H. Shi, “Multiple populations for multiple objectives: A coevolutionary technique for solving multiobjective optimization problems,” IEEE Transactions on Cybernetics, vol. 43, no. 2, pp. 445-463, April. 2013. (IF=4.943; Citation: Google Scholar 98 times, SCI 58 times)

[4]. Y. H. Li, Zhi-Hui Zhan(Corresponding Author), S. Lin, J. Zhang, and X. N. Luo, “Competitive and cooperative particle swarm optimization with information sharing mechanism for global optimization problems,” Information Sciences, vol. 293, no. 1, pp. 370-382, 2015. (IF=3.364; Citation: Google Scholar 47 times, SCI 27 times)

[5]. W. Chen, J. Zhang, Y. Lin, N. Chen, Zhi-Hui Zhan, H. Chang, Y. Li, and Y. H. Shi, “Particle swarm optimization with an aging leader and challengers,” IEEE Transactions on Evolutionary Computation, vol. 17, no. 2, pp. 241-258, April. 2013. (IF=5.908; Citation: Google Scholar 130 times, SCI 41 times)

Other Journal Papers [6]. X. F. Liu, Zhi-Hui Zhan(Corresponding Author), D. Deng, Y. Li, T. L. Gu, and J. Zhang, “An

energy efficient ant colony system for virtual machine placement in cloud computing,” IEEE Transactions on Evolutionary Computation, DOI: 10.1109/TEVC.2016.2623803. 2016. (IF=5.908)

[7]. Y. Li, Y. Zhou, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “A primary theoretical study on decomposition-based multiobjective evolutionary algorithms,” IEEE Trans. on Evolutionary Computation, vol. 20, no. 4, pp. 563-576, Aug. 2016. (IF=5.908)

[8]. Q. Lin, J. Chen, Zhi-Hui Zhan, W. Chen, C. Coello Coello, Y. Yin, C. Lim, and J. Zhang, “A hybrid evolutionary immune algorithm for multiobjective optimization problems,” IEEE Transactions on Evolutionary Computation, vol. 20, no. 5, pp. 711-729, Oct. 2016. (IF=5.908)

[9]. X. Zhang, J. Zhang, Y. Gong, Zhi-Hui Zhan, W. Chen, and Y. Li, “Kuhn-munkres parallel genetic algorithm for the set cover problem and its application to large-scale wireless sensor networks,” IEEE Transactions on Evolutionary Computation, vol. 20, no. 5, pp. 695-710, Oct. 2016. (IF=5.908)

[10]. Y. J. Gong, J. Zhang, H. Chung, W. N. Chen, Zhi-Hui Zhan, Y. Li, and Y. H. Shi, “An efficient resource allocation scheme using particle swarm optimization,” IEEE Transactions on Evolutionary Computation, vol. 16, no. 6, pp. 801-816, Dec. 2012. (IF=5.908; Citation: Google Scholar 34 times, SCI 16 times)

[11]. Y. L. Li, Zhi-Hui Zhan(Corresponding Author), Y. J. Gong, J. Zhang, Y. Li, and Q. Li, “Fast micro-differential evolution for topological active net optimization,” IEEE Transactions on Cybernetics, vol. 46, no. 6, pp. 1411-1423, Jun. 2016. (IF=4.943; Citation: Google Scholar 3 times)

[12]. Y. L. Li, Zhi-Hui Zhan(Corresponding Author), Y. J. Gong, W. N. Chen, J. Zhang, and Y. Li, “Differential evolution with an evolution path: A DEEP evolutionary algorithm,” IEEE Transactions on Cybernetics, vol. 45, no. 9, pp. 1798-1810, Sept. 2015. (IF=4.943; Citation: Google Scholar 13 times, SCI 1 times)

[13]. N. Chen, W. N. Chen, Y. J. Gong, Zhi-Hui Zhan, J. Zhang, Y. Li, and Y. S. Tan, “An evolutionary algorithm with double-level archives for multiobjective optimization,” IEEE Transactions on Cybernetics, vol. 45, no. 9, pp. 1851-1863, Sept. 2015. (IF=4.943; Citation: Google Scholar 8 times, SCI 1 times)

[14]. W. J. Yu, M. Shen, W. N. Chen, Zhi-Hui Zhan, Y. J. Gong, Y. Lin, O. Liu, and J. Zhang, “Differential evolution with two-level parameter adaptation,” IEEE Transactions on Cybernetics, vol. 44, no. 7, pp. 1080-1099, Jul. 2014. (IF=4.943; Citation: Google Scholar 31 times, SCI 12 times)

[15]. Zhi-Hui Zhan, X. Liu, H. Zhang, Z. Yu, J. Weng, Y. Li, T. Gu, and J. Zhang, “Cloudde: A heterogeneous differential evolution algorithm and its distributed cloud version,” IEEE Transactions on Parallel and Distributed Systems, vol. 28, no. 3, pp. 704-716, March. 2017. (IF=2.661)

[16]. Zhi-Hui Zhan, J. Zhang, Y. Li, O. Liu, S. K. Kwok, W. H. Ip, and O. Kaynak, “An efficient ant colony system based on receding horizon control for the aircraft arrival sequencing and scheduling problem,” IEEE Transactions on Intelligent Transportation Systems, vol. 11, no. 2, pp. 399-412, Jun. 2010. (IF=2.534; Citation: Google Scholar 78 times, SCI 33 times)

[17]. M. Shen, Zhi-Hui Zhan(Corresponding Author), W. N. Chen, Y. J. Gong, J. Zhang, and Y. Li, “Bi-velocity discrete particle swarm optimization and its application to multicast routing problem in communication networks,” IEEE Transactions on Industrial Electronics, vol. 61, no. 12, pp. 7141-7151, Dec. 2014. (IF=6.383; Citation: Google Scholar 33 times, SCI 11 times)

[18]. Y. J. Gong, M. Shen, J. Zhang, O. Kaynak, W. N. Chen, and Zhi-Hui Zhan, “Optimizing RFID network planning by using a particle swarm optimization algorithm with redundant reader elimination,” IEEE Transactions on Industrial Informatics, vol. 8, no. 4, pp. 900-912, Nov. 2012. (IF=4.708; Citation: Google Scholar 47 times, SCI 27 times)

[19]. Zhi-Hui Zhan, X. F. Liu, Y. J. Gong, J. Zhang, H. S. H. Chung, and Y. Li, “Cloud computing resource scheduling and a survey of its evolutionary approaches,” ACM Computing Surveys, vol. 47, no. 4, Article 63, pp. 1-33, Jul. 2015. (IF=5.243; Citation: Google Scholar 10 times, SCI 1 times)

[20]. Q. Liu, W. Wei, H. Yuan, Zhi-Hui Zhan(Corresponding Author), and Y. Li, “Topology selection for particle swarm optimization,” Information Sciences, vol. 363, no. 1, pp. 154-173, Oct. 2016. (IF=3.364)

[21]. J. Zhang, Zhi-Hui Zhan, Y. Lin, N. Chen, Y. J. Gong, J. H. Zhong, H. S. H. Chung, Y. Li, and Y. H. Shi, “Evolutionary computation meets machine learning: A survey,” IEEE Computational Intelligence Magazine, vol. 6, no. 4, pp. 68-75, Nov. 2011. (IF=3.647; Citation: Google Scholar 77 times, SCI 41 times)

[22]. Y. Gong, W. Chen, Zhi-Hui Zhan, J. Zhang, Y. Li, Q. Zhang, and J. Li, “Distributed evolutionary algorithms and their models: A survey of the state-of-the-art,” Applied Soft Computing, vol. 34, pp. 286-300, Sept. 2015. (IF=2.857; Citation: Google Scholar 10 times, SCI 2 times)

[23]. W. Yu, Zhi-Hui Zhan, and J. Zhang, “Artificial bee colony algorithm with an adaptive greedy position update strategy,” Soft Computing, DOI:10.1007/s00500-016-2334-4. 2016. (IF=1.630)

Selected Conference Papers (First/Corresponding Author, Most of Them Are

CEC/GECCO/SSCI Papers)

[1]. Zhi-Hui Zhan, Z. J. Wang, Y. Lin, and J. Zhang, “Adaptive radius species-based particle swarm optimization for multimodal optimization problems,” in Proc. IEEE Congr. Evol. Comput. (CEC 2016), Vancouver, Canada, Jul. 2016, pp. 2043-2048.

[2]. Z. J. Wang, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “Orthogonal learning particle swarm optimization with variable relocation for dynamic optimization,” in Proc. IEEE Congr. Evol. Comput. (CEC 2016), Vancouver, Canada, Jul. 2016, pp. 594-600.

[3]. X. F. Liu, Zhi-Hui Zhan(Corresponding Author), J. H. Lin, and J. Zhang, “Parallel differential evolution on distributed computational resources for power electronic circuit optimization,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2016), 2016, pp. 117-118.

[4]. Z. J. Wang, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “Parallel multi-strategy evolutionary algorithm using massage passing interface for many-objective optimization,” in Proc. IEEE Symposium Series on Computational Intelligence (SSCI 2016), Athens, Greece, Dec. 2016, pp. 1-8.

[5]. Z. G. Chen, Zhi-Hui Zhan(Corresponding Author), W. Shi, W. N. Chen, and J. Zhang, “When neural network computation meets evolutionary computation: A survey,” in Proc. International Symposium on Neural Networks (ISNN 2016), Saint Petersburg, Russia, Jul. 2016, pp. 603-612.

[6]. Y. F. Li, Zhi-Hui Zhan(Corresponding Author), Y. Lin, and J. Zhang, “Comparisons study of APSO OLPSO and CLPSO on CEC2005 and CEC2014 test suits,” in Proc. IEEE Congr. Evol. Comput. (CEC 2015), Sendai, Japan, 2015, pp. 3179-3185.

[7]. Z. G. Chen, K. J. Du, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “Deadline constrained cloud computing resources scheduling for cost optimization based on dynamic objective genetic algorithm,” in Proc. IEEE Congr. Evol. Comput. (CEC 2015), Sendai, Japan, 2015, pp. 708-714.

[8]. H. H. Li, Y. W. Fu, Zhi-Hui Zhan(Corresponding Author), and J. J. Li, “Renumber strategy enhanced particle swarm optimization for cloud computing resource scheduling,” in Proc. IEEE Congr. Evol. Comput. (CEC 2015), Sendai, Japan, 2015, pp. 870-876.

[9]. X. F. Liu, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “Dichotomy guided based parameter adaptation for differential evolution,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2015), Madrid, Spain, Jul. 2015, pp. 289-296.

[10]. H. H. Li, Z. G. Chen, Zhi-Hui Zhan(Corresponding Author), K. J. Du, and J. Zhang, “Renumber coevolutionary multiswarm particle swarm optimization for multi-objective workflow scheduling on cloud computing environment,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2015), Madrid, Spain, Jul. 2015, pp. 1419-1420.

[11]. Z. J. Wang, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “An improved method for comprehensive learning particle swarm optimization,” in Proc. IEEE Symposium Series on Computational Intelligence (SSCI 2015), Cape Town, South Africa, Dec. 2015, pp. 218-225.

[12]. Zhi-Hui Zhan and J. Zhang, “Differential evolution for power electronic circuit optimization,” in Proc. Conf. Technologies and Applications of Artificial Intelligence (TAAI 2015), Tainan, Taiwan, Nov. 2015, pp. 158-163.

[13]. Z. G. Chen, Zhi-Hui Zhan(Corresponding Author), H. H. Li, K. J. Du, J. H. Zhong, Y. W. Foo, Y. Li, and J. Zhang, “Deadline constrained cloud computing resources scheduling through an ant colony system approach,” in Proc. Int. Conf. Cloud Computing Research and Innovation (ICCCRI 2015), Singapore, Oct. 2015, pp. 112-119.

[14]. Zhi-Hui Zhan, J. J. Li, and J. Zhang, “Adaptive particle swarm optimization with variable relocation for dynamic optimization problems,” in Proc. IEEE Congr. Evol. Comput. (CEC 2014), Beijing, China, Jul. 2014, pp. 1565-1570.

[15]. X. F. Liu and Zhi-Hui Zhan(Corresponding Author), “Energy aware virtual machine placement scheduling in cloud computing based on ant colony optimization approach,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2014), Vancouver, Canada, Jul., 2014, pp. 41-47.

[16]. G. W Zhang and Zhi-Hui Zhan(Corresponding Author), “A normalization group brain storm optimization for power electronic circuit optimization,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2014), Vancouver, Canada, Jul., 2014, pp. 183-184.

[17]. Zhi-Hui Zhan, G. Y. Zhang, Y. J. Gong, and J. Zhang, “Load balance aware genetic algorithm for task scheduling in cloud computing,” in Proc. Simulated Evolution And Learning (SEAL 2014), Dec. 2014, pp. 644-655.

[18]. Meng-Dan Zhang, Zhi-Hui Zhan(Corresponding Author), J. J. Li, and J. Zhang, “Tournament selection based artificial bee colony algorithm with elitist strategy,” in Proc. Conf. Technologies and Applications of Artificial Intelligence (TAAI 2014), Taiwan, Nov. 2014, pp. 387-396.

[19]. Guang-Wei Zhang, Zhi-Hui Zhan(Corresponding Author), K. J. Du, Y. Lin, W. N. Chen, J. J. Li, and J. Zhang, “Parallel particle swarm optimization using message passing interface,” in Proc. The 18th Asia Pacific Symposium on Intelligent and Evolutionary Systems (IES 2014), Singapore, Nov. 2014, pp. 55-64.

[20]. Y. L. Li, and Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “Differential evolution enhanced with evolution path vector,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2013), Jul., 2013, pp. 123-124.

[21]. Zhi-Hui Zhan, W. N. Chen, Y. Lin, Y. J. Gong, Y. L. Li, and J. Zhang, “Parameter investigation in brain storm optimization,” in Proc. IEEE Symposium Series on Computational Intelligence (SSCI 2013), Singapore, April. 2013, pp. 103-110.

[22]. Zhi-Hui Zhan, J. Zhang, Y. H. Shi, and H. L. Liu, “A modified brain storm optimization,” in Proc. IEEE Congr. Evol. Comput. (CEC 2012), Brisbane, Australia, Jun. 2012, pp. 1-8.

[23]. Zhi-Hui Zhan and J. Zhang “Enhance differential evolution with random walk,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2012), Philadelphia, America, Jul. 2012, pp. 1513-1514.

[24]. Zhi-Hui Zhan, K. J. Du, J. Zhang, and J. Xiao, “Extended binary particle swarm optimization approach for disjoint set covers problem in wireless sensor networks,” in Proc. Conf. Technologies and Applications of

Artificial Intelligence (TAAI 2012), Tainan, Taiwan, 2012. pp. 327-331. [25]. Zhi-Hui Zhan n, and J. Zhang, “Orthogonal learning particle swarm optimization for power electronic

circuit optimization with free search range,” in Proc. IEEE Congr. Evol. Comput. (CEC 2011), New Orleans, Jun. 2011, pp. 2563-2570.

[26]. Zhi-Hui Zhan n and J. Zhang, “Co-evolutionary differential evolution with dynamic population size and adaptive migration strategy,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2011), Dublin, Ireland, Jul., 2011, pp. 211-212.

[27]. Zhi-Hui Zhan and J. Zhang, “Self-adaptive differential evolution based on PSO learning strategy,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2010), Portland, America, Jul., 2010, pp. 39-46.

[28]. Zhi-Hui Zhan and J. Zhang, “A parallel particle swarm optimization approach for multiobjective optimization problems,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2010), Portland, America, Jul., 2010, pp. 81-82.

[29]. Zhi-Hui Zhan, J. Zhang, and Z. Fan, “Solving the optimal coverage problem in wireless sensor networks using evolutionary computation algorithms,” in Proc. Simulated Evolution And Learning (SEAL 2010), LNCS 6457, pp. 166–176, 2010.

[30]. Zhi-Hui Zhan, J. Zhang, and Y. H. Shi, “Experimental study on PSO diversity,” in Proc. 3rd Int. Workshop on Advanced Computational Intelligence (IWACI 2010), Suzhou, China, Aug. 2010, pp. 310-317.

[31]. Zhi-Hui Zhan, X. L. Feng, Y. J. Gong, and J. Zhang, “Solving the flight frequency programming problem with particle swarm optimization,” in Proc. IEEE Congr. Evol. Comput. (CEC 2009), Trondheim, Norway, May. 2009, pp. 1383-1390.

[32]. Zhi-Hui Zhan, J. Zhang, and R. Z. Huang, “Particle swarm optimization with information share mechanism,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2009), Montréal, Canada, Jul., 2009, pp. 1761-1762.

[33]. Zhi-Hui Zhan and J. Zhang, “Parallel particle swarm optimization with adaptive asynchronous migration strategy,” in Proc. The 9th Int. Conf. on Algorithms and Architectures for Parallel Processing (ICA3PP), Taipei, Taiwan, Jun., 2009, pp. 490-501.

[34]. Zhi-Hui Zhan and J. Zhang, “Discrete particle swarm optimization for multiple destination routing problems,” in Proc. EvoWorkshops 2009, LNCS 5484, April. 2009, pp. 117–122.

[35]. Zhi-Hui Zhan, J. Xiao, J. Zhang, and W. N. Chen, “Adaptive control of acceleration coefficients for particle swarm optimization based on clustering analysis,” in Proc. IEEE Congr. Evol. Comput. (CEC 2007), Singapore, Sept. 2007, pp. 3276-3282.

Authorized Patents

[1]. J. Zhang, Zhi-Hui Zhan, and T. Huang, Multicast Approach Based on Particle Swarm Optimization, Patent No. ZL200810220650.1.

Summary of Key Publications in SCI Journals

Journal Names 5-Years IF 2016’s IF 5-Year IF Rank in JCR Category PapersIEEE Trans. Evol. Comput. 6.897 5.908 Computer Science – Theory & Method 1/105 7 IEEE Trans. SMC. Part B (CYB) 4.978 4.943 Computer Science – Cybernetics 1/22 6 IEEE Trans. Ind. Electron. 5.985 6.383 Automation & Control System 1/59 1 IEEE Trans. Intell. Transp.Syst. 3.155 2.534 Transportation Science & Technology 6/33 1 IEEE Trans. Ind. Informatics 4.880 4.708 Automation & Control System 3/59 1 IEEE Trans. Paral. Distr. Syst. 2.749 2.661 Computer Science – Theory & Method 11/105 1 IEEE Comput. Intell. Mag. 3.483 3.647 Computer Science – Artificial Intelligence 18/130 1 ACM Computing Surveys 6.559 5.243 Computer Science – Theory & Method 2/105 1 Information Sciences 3.683 3.364 Computer Science – Information Systems 10/144 2 Applied Soft Computing 3.288 2.857 Computer Science – Interdiscip. Appl. 14/104 1 Soft Computing 1.732 1.630 Computer Science – Artificial Intelligence 57/130 1 Total 23

Part II: Referee Letters

Derong Liu Professor

Phone (312) 355-4475

Fax (312) 996-6465

[email protected] www.ece.uic.edu

March 20, 2017

Re: IEEE CIS Outstanding Ph.D. Dissertation Award

Dear Chair and Members of the IEEE CIS Award Committee:

I am glad to write this letter to support Dr. Zhi-Hui Zhan for the 2017 IEEE CIS Outstanding Ph.D.

Dissertation Award. The dissertation “Research into Machine Learning Aided Particle Swarm Optimization and

Its Engineering Application” deals with challenges on introducing machine learning (ML) techniques into

evolutionary computation (EC) algorithms, specially the particle swarm optimization (PSO), to enhance the

performance. The topic of the combination of ML and EC is interesting and is a new idea for EC algorithm

design. This is a nice dissertation contributes to PSO development, by mainly addressing the following three

challenges:

1) To address the challenge that parameters are sensitive to different problems in PSO, this dissertation

designs an adaptive PSO (APSO) to dynamically adjust parameters based on statistical analysis

methods in ML. This work makes PSO less sensitive to the parameter settings and more efficient in a

wider range of applications;

2) To address the challenge in global search capacity of PSO, this dissertation proposes an orthogonal

learning PSO (OLPSO) based on orthogonal experimental design and orthogonal prediction techniques

to construct a promising exemplar to guide the swarm evolution. This work enhance the global search

capacity and speed up the algorithm convergence;

3) To address the fitness assignment problem in multiobjective optimization, this dissertation develops a

co-evolutionary multi-swarm PSO which is inspired by the ensemble learning in ML to obtain well

distribution solutions along the Pareto front.

In addition to general algorithm design studies, this dissertation also extends the ML aided PSO to

engineering application problems. The successful application on power electronic circuit optimization

demonstrates the efficiency and effectiveness of ML aided PSO.

The main content of the dissertation has been published in 13 most related papers, including 9 papers in

IEEE Transactions, 1 paper in IEEE CIM magazine, 1 paper in Elsevier Information Science, and 2 papers in

the top conferences CEC and GECCO. Therefore, the significance of the research is evident as shown by these

high impact journal and conference publications that have been produced from this PhD dissertation. Moreover,

3 of these papers are ESI Highly Cited papers, such as the APSO in IEEE Transactions on Systems, Man, and

Cybernetics - Part B, and the OLPSO in IEEE Transactions on Evolutionary Computation.

Significantly, based on the related research work in this dissertation, Dr. Zhan has published 7 high

quality papers in TEVC in recent 5 years (from 2011), and is listed as one of the top authors according to

the number of papers. This impresses me greatly that a young scholar can obtain such huge achievements

during his Ph.D. study and in his early career after the Ph.D. degree. Due to the excellent work, this Ph.D.

dissertation was awarded the China Computer Federation (CCF) Outstanding Dissertation in 2014,

which is evaluated every year covering the Ph.D. Dissertations in the last 2 years all over China in all fields

related to computer science, such as artificial intelligence, networks, database, security, vision, software

engineering, and architecture.

By considering the above exceptional excellence and contributions of Dr. Zhi-Hui Zhan’s dissertation, I

strongly support the nomination of Dr Zhi-Hui Zhan for the IEEE CIS Outstanding PhD Dissertation Award.

Sincerely, Derong Liu Professor [email protected] http://www.ece.uic.edu/~derong/

mailto:[email protected]

http://www.ece.uic.edu/~derong/

CENTRO DE INVESTIGACION Y DE ESTUDIOS AVANZADOS DEL I.P.N.

Av. Instituto Politécnico Nacional # 2508 Col. San Pedro Zacatenco México, D.F. C.P. 07360 Tel. 50-61-38-00 Fax: 57-47-38-02

Mexico City, Mexico, April 11th, 2017 To whom it may concern: This letter is to express my support for the nomination of the PhD thesis entitled “Research into Machine Learning Aided Particle Swarm Optimization and Its Engineering Application” written by Dr. Zhi-Hui Zhan, to the IEEE CIS Outstanding Ph.D. Dissertation Award. The PhD thesis of Dr. Zhan deals with a combination of machine learning techniques and evolutionary computation algorithms. This thesis contains 4 main contributions: 1) It proposes an adaptive particle swarm optimization (APSO) algorithm which uses statistical analysis to estimate the current state of the search during its execution. This scheme is used to control the parameters of APSO in an automated way when applying it to different types of problems, thus significantly improving the efficiency and generality of this particle swarm optimizer (PSO). The main paper derived from this work was published in the IEEE Transactions on Systems, Man and Cybernetics Part B. This paper is the second most highly cited in this journal in the last 10 years. 2) It proposes an orthogonal learning PSO (OLPSO), which uses orthogonal experimental design (OED) to discover useful search information in the currently available solutions, in order to enhance the global search ability of the algorithm. Using orthogonal prediction techniques, the author proposes a novel orthogonal learning strategy for the particles to construct a promising sample that is used to properly guide the search. As a result, OLPSO can attain a fast convergence while having a strong global search ability. The main paper derived from this work was published in the IEEE Transactions on Evolutionary Computation. It is worth noting that this paper has obtained 351 citations in Google Scholar (200 in the ISI Web of Science). 3) It proposes a co-evolutionary multi-swarm PSO (CMPSO). This thesis proposes a novel optimization framework that adopts multiple populations for multiple objectives. This proposal is based on the idea of having multiple learners. Within this framework, CMPSO avoids the fitness assignment problem that arises when considering all the objectives together, while also producing a better search in different areas of the Pareto front by using the guidance of each objective. The main paper derived from this work was published in the IEEE Transactions on Cybernetics. 4) OLPSO is applied to the design of power electronic circuits. In this application, a novel contribution is that the author introduces a free search range for the electronic components. The main paper derived from this work was published in the IEEE Congress on Evolutionary Computation.

CENTRO DE INVESTIGACION Y DE ESTUDIOS AVANZADOS DEL I.P.N.

Av. Instituto Politécnico Nacional # 2508 Col. San Pedro Zacatenco México, D.F. C.P. 07360 Tel. 50-61-38-00 Fax: 57-47-38-02

It is worth emphasizing that a total of 13 papers were derived from this PhD thesis: 3 in the IEEE Transactions on Evolutionary Computation, 4 in the IEEE Transactions on Systems, Man, and Cybernetics Part B (and the IEEE Transactions on Cybernetics), 1 in the IEEE Transactions on Industrial Electronics, 1 the IEEE Transactions on Parallel and Distributed Systems, 1 in the IEEE Computational Intelligence Magazine, 1 in Information Sciences, 1 in the IEEE Congress on Evolutionary Computation and 1 in the Genetic and Evolutionary Computation Conference. Based on the previous, and considering that I am convinced that this PhD thesis makes a valuable contribution to the computational intelligence field, I strongly support it for the IEEE CIS Outstanding Ph.D. Dissertation Award. Please don’t hesitate to contact me in case you have any questions about this matter. Sincerely,

Dr. Carlos A. Coello Coello IEEE Fellow Professor with Distinction (Investigador Cinvestav 3F) Computer Science Department CINVESTAV-IPN Av. IPN No. 2508 Col. San Pedro Zacatenco México, D.F. 07360, México Tel. +52 (55) 5747 3800 x 6564 Fax +52 (55) 5747 3757 email: [email protected] URL: http://delta.cs.cinvestav.mx/~ccoello

City Unìversityof Hong Kong

www.cityu.edu.hk

A Ã)ËnÈEì*zß&Tat Chee Avenue, Kowloon, Hong Kong

T (8s2)3442 8s80 F (852)34420503

E [email protected]Ëüñ7\et* â1ff ÊËè4Professional. cr€at¡vetor The world €[Ér+H.ã

Computer Science

Reference Letter to Support Dr. Zhi-Hui Zhan for IEEE CIS Outstanding

Ph. D. Dissertation Award

April ll,2017

To theAward Committee:

I am writing this letter to support the Ph. D. dissertation by Dr. Zhi-Hui Zhan on

"Research into Machine Learning Aided Particle Swarm Optimization and Its

Engineering Application" for the 2017 IEEE CIS Outstanding Ph.D. Dissertation

Award. This is a nice dissertation with general purpose algorithm development studies

and its engineering application studies. More specifically, three new machine leaning(ML) aided PSO algorithms have been proposed in this dissertation. An adaptive PSO

algorithm aided by the statistics methods in ML, an orthogonal learning PSO

algorithm aided by the orthogonal experiments design technique in ML, and a

co-evolutionary multi-swarm PSO algorithm inspired by the ensemble learning

technique in ML. By studying these ML aid PSO variants and their applications, this

dissertation has conducted a nice combination of evolutionary computation (EC) and

ML, which are two significant fields in computer science.

The main content of the dissertation has been published in 13 most related papers

include l0 IEEE Transactions papers, and 4 of them are ESI Highly Cited Papers,

as follows:

[1]. Zhi-Hui Zhan, J. Zhatg, Y. Li, and H, Chung, "Adaptive particle swarm optimization," IEEETrflnssctions on Systems, Man, ønd Cybernetics - Pørt B, vol. 39, no. 6, pp. 1362-l38l,Dec.

2009. [Related to Chapter 2: on the adaptive PSO]

o Google Scholar Citation 1028 times, SCI Citation 511 times

l2l,Zhi- ui Zhan, J. Zhang, Y, Li, and Y. H. Shi, "Orthogonal learning parlicle swarm

optimization," IEEE Trunssctions on Evolulionary Computøtion, , vol. 15, no, 6, pp.

832-847 , Dec. 2011. [Related to Chapter 3: on the orthogonal learning PSO]


[3].Zhi-Hui Zhan, J. Li, J. Cao, J. Zhang, H, Chung, and Y. H. Shi, "Multiple populations for

multiple objectives: A coevolutionary technique for solving multiobjective optimization

problems," IEEE Transuctions on Cybernetics, vol.43, no. 2, pp. 445-463, April. 2013.

[Related to Chapter 4: : Propose the co-evolutionary multiswarm PSO for MOP]


I4l.Y. H. Li, Zhi-Hui Zhan (Correspondine Author), S. Lin, J. Zhang, and X, N. Luo,

"Competitive and cooperative particle swarm optimization with information sharing

mechanism for global optimization problems," InformalÍon Sciences, vol. 293, no. l, pp,

370-382,2015, [Related to Chapter 3: Extension of the orthogonal learning strategy to

competitive and cooperative strategyl


The rich productions of this dissertation impress me a lot. Moreover, the high

citations of these papers in both ISI Web of Science and Google Scholar clearly show

their high impacts and their large contributions to the freld of EC. The adaptive PSO

work [1] published in the IEEE TSMCB (the journal that I am current serving as EiC)

has even reached over 1000 citations in Google Scholar and over 500 citations in ISI.

Significantly, it is the second hishest citation paner amons all the papers durinsthe last 10 years (since 2006) that published in TSMCB and TCYB. Moreover, the

orthogonal learning PSO work [2] is the third hishest citation paper amons all thepapers durinq the last 5 years lsince 20Ll) that published in TEVC, which is the

Ieadins iournal in our CIS community.

Based on the above mentioned contributions of Dr. Zhi-}jui Zhan's dissertation,

from both the aspects of large number of published IEEE Transactions papers and

high citation of the papers, I strongly recommend his Ph. D. dissertation for the 2017

IEEE CIS Outstanding Ph. D. Dissertation Award.

Sincerely;

¿t/

Jun Wang, PhD

Chair Professor of Computational Intelligence

Reference Letter for 2017 IEEE CIS Outstanding Ph. D. Dissertation

Award Application

To: IEEE CIS Outstanding PhD Dissertation Award Committee:

I am writing to provide a reference letter to support Dr. Zhi-Hui Zhan for the IEEE CIS Outstanding PhD Dissertation Award, for his PhD dissertation “Research into Machine Learning Aided Particle Swarm Optimization and Its Engineering Application”.

His dissertation is related both the evolutionary computation (EC) field and the machine learning (ML) field. The population-based search and iterative-based evolution of EC algorithms generate mass of search data during the evolutionary process. This makes it possible and promising to introduce ML techniques into EC algorithms to enhance the algorithm performance. Therefore, this dissertation focuses on this and makes significant contributions to ML aided EC research and applications.

Specially, this dissertation proposes three novel particle swarm optimization (PSO) variants aided by different ML techniques. Firstly, designing an adaptive PSO (APSO) based on statistical analysis technique in ML to relieve the sensitivity of parameters in PSO. Secondly, proposing orthogonal learning PSO (OLPSO) by using the orthogonal experimental design technique in ML to enhance the rapid global search capability. Thirdly, developing a co-evolutionary multi-swarm PSO (CMPSO) by introducing the ensemble learning technique in ML to enhance the search efficiency of PSO in multi-objective problems. The proposed algorithms in the dissertation demonstrate that the ML aided PSO variants can significantly enhance the PSO’s ability in handling optimization problems. Furthermore, this dissertation extends the application of OLPSO to the challenging power electronic circuit design problem and demonstrates the effectiveness and efficiency of OLPSO in engineering application.

The novelty and significance of this dissertation have been demonstrated by 13 key publications in reputable journals and conference proceedings generated by the research reported in the dissertation. Among these 13 papers, 7 papers are published in the IEEE Transactions on Evolutionary Computation and the IEEE Transactions on Cybernetics (previous SMC Part B), which are both reputable journals in CIS and are with very low acceptance rates.

Not only the quantity is impressive, the quality of the papers is also very impressive. These papers have been cited by more than 1500 times in Google Scholar and about 1000 times in SCI. Moreover, 4 papers of them are ESI Highly Cited Papers. The OLPSO was even the ESI Hot Paper in Feb. 2014, being one of

the only 62 ESI Hot Papers in Computer Science field all over the world. Significantly, the APSO, OLPSO, and CMPSO research topics related to Chapters 2, 3, and 4, respectively, in this dissertation are listed as the ESI research front, showing their broadly impact in the EC and related communities.

It is absolutely that the outcomes of Dr. Zhi-Hui Zhan’s dissertation are impressive, not only from the large number of related high quality papers published in the CIS reputable journals and conferences, but also from the significant impact of these works in the EC field and even across different fields in computer science.

Based on the above, I strongly believe that this dissertation is with sufficient and prior quality for the IEEE CIS Outstanding PhD Dissertation Award. Therefore, I show my strongest recommendation to support this dissertation for this award.

Best regards,

C. L. Philip Chen, Ph.D., FIEEE, FAAAS

EiC, IEEE Transactions on Systems, Man, and Cybernetics: Systems

Dean, Faculty of Science and Technology, University of Macau

Reference Letter for IEEE CIS Outstanding Ph. D. Dissertation Award

To the CIS Award Committee:

It is my pleasure to recommend Dr. Zhi-Hui Zhan for applying the IEEE CIS Outstanding Ph. D. Dissertation Award. The title of the dissertation is “Research into Machine Learning Aided Particle Swarm Optimization and Its Engineering Application” which relates to using machine learning (ML) techniques to enhance the search ability of evolutionary computation (EC) algorithms, e.g., the particle swarm optimization (PSO). This dissertation has conducted a systematic research on this cross field. Specially, the dissertation proposes 3 novel ML aided PSO variants in Chapters 2, 3, and 4 respectively, and extends the algorithm to engineering application.

Firstly, an adaptive PSO (APSO) is proposed in Chapter 2, by using the ML statistical methods to analyze the population distribution information, so as to adaptive control the PSO parameters and operators. The key publication of this Chapter is:

[1]. Zhi-Hui Zhan (First Author), “Adaptive particle swarm optimization,” IEEE Transactions on Systems, Man, and Cybernetics--Part B, vol. 39, no. 6, pp. 1362-1381, Dec. 2009.

ESI Highly Cited Paper, ESI Research Front Google Scholar Citation 1026 times, SCI Citation 511 times The Top 2 cited paper of this journal in recent 10 years, since 2006

Secondly, an orthogonal learning PSO (OLPSO) is proposed in Chapter 3, by using the orthogonal experimental design (OED) method in ML filed to conduct more promising learning guidance for the particle, so as to enhance the global search ability. The key publications of this Chapter are:

[2]. Zhi-Hui Zhan (First Author), “Orthogonal learning particle swarm optimization,” IEEE Transactions on Evolutionary Computation, , vol. 15, no. 6, pp. 832-847, Dec. 2011.

ESI Hot Paper, ESI Highly Cited Paper, ESI Research Front Google Scholar Citation 351 times, SCI Citation 196 times The Top 3 cited paper of this journal in recent 5 years, since 2011

[3]. Zhi-Hui Zhan (Corresponding Author), “Competitive and cooperative particle swarm optimization with

information sharing mechanism for global optimization problems,” Information Sciences, vol. 293, no. 1, pp. 370-382, 2015.

ESI Highly Cited Paper, ESI Research Front Google Scholar Citation 47 times, SCI Citation 27 times

Thirdly, a co-evolutionary multi-swarm PSO (CMPSO) is proposed in Chapter 4, by designing a novel optimization framework for multi-objective optimization problem (MOP). The new framework is “multiple populations for multiple objectives (MPMO)”, inspired by the idea of ensemble learning in the ML filed. The key publication of this Chapter is:

[4]. Zhi-Hui Zhan (First Author), “Multiple populations for multiple objectives: A coevolutionary technique

for solving multiobjective optimization problems,” IEEE Transactions on Cybernetics, vol. 43, no. 2, pp. 445-463, April. 2013.

ESI Highly Cited Paper, ESI Research Front Google Scholar Citation 98 times, SCI Citation 58 times

It impresses me greatly that this dissertation has produced so many high quality papers. All the key papers in all the PSO variants have caused great attentions from the research community. All the above 3 research topics are listed as ESI Research Front and all these 4 papers are ESI Highly Cited Papers.

Including the above, this dissertation has produced 13 most related papers, and 9 of them are published in IEEE Transactions, like the leading journals in our CIS community, e.g., the IEEE TEVC and IEEE TCYB (SMCB).

Based on the above-mentioned exceptional excellence of Dr. Zhi-Hui Zhan’s dissertation, I strongly support his Ph. D. dissertation for the 2017 IEEE CIS Outstanding Ph. D. Dissertation Award.

Sincerely yours, Chin-Teng(CT) Lin, IEEE Fellow Distinguished Professor, UTS

Part III: Ph. D. Dissertation

Research into Machine Learning Aided Particle Swarm

Optimization and Its Engineering Application

Major： Computer Science

Doctorate Applicant： Zhi-Hui Zhan Supervisor： Prof. Jun Zhang

Supervisory Committee Members： Chair: South China University of Techolongy, China

Prof. Guo-Qiang Hang

Member: University of Surrey, UK Prof. Yao-Chu Jin

South China Normal University, China

Prof. Yong Tang Sun Yat-sen University, China

Ptof. Xiao-Nan Luo Sun Yat-sen University, China

Prof. Xiao-La Lin

Acknowledgments

I would take this opportunity to express my sincere gratitude to those who have helped

me and supported me during my perusing of the Ph. D degree.

First of all, I would like to thank my supervisor Prof. Jun Zhang. He has leaded me into

the academic world of evolutionary computation and particle swarm optimization. During my

Ph. D. study, he always inspired me and guided me to make deep researches that can catch

the progress of international developments. Without his encouragement and supervisor, all

these work would not have been possible.

I would also like to thank Prof. Henry Chung, Prof. Yun Li, Prof. Y. H. Shi, and Prof. K.

C. Tan. They are all the seniors in the academic community and have given me great help. It

is my great luck and honour that I can learn much from them during their visit to our lab.

They show great patience.to discuss the problems that I faced during the researches, and

provide lots of creative valuable comments and suggestions to my research work.

Thanks also to the friends in the lab, especially Jing-Hui Zhong, Xiao-Min Hu,

Wei-Neng Chen, Ying Lin, Wei-Jie Yu, Yuan-Long, Li, and Yue-Jiao Gong. They have

worked with me days and nights, together happy and together tears. Moreover, the

discussions with them have greatly enhanced the quality of my research work.

At last, but also the most importantly, I wish to thank my parents and my wife for their

unconditional love, support, and encouragement.

Declaration of Thesis Originality

I solemnly declare: the submitted thesis is the achievemets of my independent

research work under the conduct of my instructor. Except for the context annotated

references in the thesis, this thesis does not contain archivements that any other personal

or team have published or written. The persons or teams who make important

constribution to my research work have already been marked in the thesis. I am fully

aware that I will take the blame for the legal consequences of this statement by myself.

Signature (Author): Zhi-Hui Zhan

Date: 18 May, 2013

Permission to Quote Copyrighted Material

I fully understand the rules about dissertation reservation and use in Sun Yat-sen

University: School has the right to reserve and submit my dissertation in electronic or

printed format to the national competent authority or its designated agency. School has

the right to make few copies of my dissertation for nonprofitable purpose and allow my

dissertation to be accessed in school’s library and college reference rooms. School has the

right to index my dissertation into related databases for retrieval, and use copy, copy in a

reduced format, or other methods to save my dissertation. Signature (Author): Zhi-Hui Zhan Signature (Suipervisor): Jun Zhang Date: 18 May, 2013 Date: 18 May, 2013

Brief CV & Publications

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Work and Education 01/2016 – Now Professor

School of Comput. Sci. and Eng. South China Univ of Techno

01/2015 – 12/2015 Associate Professor School of Advanced Computing Sun Yat-sen University

07/2013 – 12/2014 Lecturer School of Info. Sci. and Technology Sun Yat-sen University

09/2009 – 06/2013 Ph. D. Computer Application Technology Sun Yat-sen University

09/2003 – 07/2007 Bachelor of Science Computer Science and Technology Sun Yat-sen University

Awards and Honors 2016, Pearl River Scholar Young Professor

2015, Elsevier Most Cited Chinese Researchers in Computer Science

2014, Elsevier Most Cited Chinese Researchers in Computer Science

2014, Natural Science Found for Distinguished Young Scholars, GD, China

2015, Pearl River New Star in Science and Technology

2014, Guangdong Province Outstanding Dissertation Award

2013, China Computer Federation (CCF) Outstanding Dissertation Award

Research Interests Computation Intelligence

Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), Genetic Algorithm (GA), Differential Evolution (DE), Brain Storm Optimization (BSO)

Cloud Computing and Big Data

Large Scale Resources Scheduling and Management, Multiobjective Optimization, Dynamic Optimization, Multimodual Optimization

Intelligent Application

Wireless Sensor Network, Scheduling and Control, Intelligent Systems


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Publications Related to the Thesis


[1]. Zhi-Hui Zhan, J. Zhang, Y. Li, and H. Chung, “Adaptive particle swarm optimization,” IEEE Transactions on Systems, Man, and Cybernetics--Part B, vol. 39, no. 6, pp. 1362-1381, Dec. 2009. [Related to Chapter 2: Propose the adaptive PSO]




[2]. Y. L. Li, Zhi-Hui Zhan (Corresponding Author), Y. J. Gong, W. N. Chen, J. Zhang, and Y. Li, “Differential evolution with an evolution path: A DEEP evolutionary algorithm,” IEEE Transactions on Cybernetics, vol. 45, no. 9, pp. 1798-1810, Sept. 2015. [Related to Chapter 2: Extend the adaptive idea to DE]



[3]. Zhi-Hui Zhan, J. Zhang, Y. Li, and Y. H. Shi, “Orthogonal learning particle swarm optimization,” IEEE Transactions on Evolutionary Computation, vol. 15, no. 6, pp. 832-847, Dec. 2011. [Related to Chapter 3: Propose the orthogonal learning PSO]

ESI Hot Paper, ESI Highly Cited Paper



[4]. Y. H. Li, Zhi-Hui Zhan (Corresponding Author), S. Lin, J. Zhang, and X. N. Luo, “Competitive and cooperative particle swarm optimization with information sharing mechanism for global optimization problems,” Information Sciences, vol. 293, no. 1, pp. 370-382, 2015. [Related to Chapter 3: Extend the orthogonal learning strategy to competitive and cooperative strategy]




[5]. Zhi-Hui Zhan, J. Li, J. Cao, J. Zhang, H. Chung, and Y. H. Shi, “Multiple populations for multiple objectives: A coevolutionary technique for solving multiobjective optimization problems,” IEEE Transactions on Cybernetics, vol. 43, no. 2, pp. 445-463, April. 2013. [Related to Chapter 4: : Propose the co-evolutionary multiswarm PSO for MOP]



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[6]. Y. L. Li, Y. R. Zhou, Zhi-Hui Zhan (Corresponding Author), and J. Zhang, “A primary theoretical study on decomposition-based multiobjective evolutionary algorithms,” IEEE Transactions on Evolutionary Computation, vol. 20, no. 4, pp. 563-576, Aug. 2016. [Related to Chapter 4: Theoretical study on multiobjective evolutionary algorithms]

[7]. H. H. Li, Z. G. Chen, Zhi-Hui Zhan (Corresponding Author), K. J. Du, and J. Zhang, “Renumber coevolutionary multiswarm particle swarm optimization for multi-objective workflow scheduling on cloud computing environment,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2015), Madrid, Spain, Jul. 2015, pp. 1419-1420. [Related to Chapter 4: Apply the CMPSO algorithm to cloud computing resources scheduling]


[8]. Zhi-Hui Zhan and J. Zhang, “Orthogonal learning particle swarm optimization for power electronic circuit optimization with free search range,” in Proc. IEEE Congr. Evol. Comput. (CEC 2011), New Orleans, Jun. 2011, pp. 2563-2570. [Related to Chapter 5: Propose to use OLPSO to solve PEC]

[9]. M. Shen, Zhi-Hui Zhan (Corresponding Author), W. N. Chen, Y. J. Gong, J. Zhang, and Y. Li, “Bi-velocity discrete particle swarm optimization and its application to multicast routing problem in communication networks,” IEEE Transactions on Industrial Electronics, vol. 61, no. 12, pp. 7141-7151, Dec. 2014. [Related to Chapter 5: Engineering application of PSO]


[10]. X. F. Liu, Zhi-Hui Zhan (Corresponding Author), and J. Zhang, “An energy efficient ant colony system for virtual machine placement in cloud computing,” IEEE Transactions on Evolutionary Computation, DOI: 10.1109/TEVC.2016.2623803. 2016. [Related to Chapter 5: Engineering application]

[11]. Y. L. Li, Zhi-Hui Zhan (Corresponding Author), Y. J. Gong, J. Zhang, Y. Li, and Q. Li, “Fast micro-differential evolution for topological active net optimization,” IEEE Transactions on Cybernetics, vol. 46, no. 6, pp. 1411-1423, Jun. 2016. [Related to Chapter 5: Engineering application]

Publications Related to Chapter 1&6

[12]. J. Zhang (Supervisor), Zhi-Hui Zhan, Y. Lin, N. Chen, Y. J. Gong, J. H. Zhong, H. S. H. Chung, Y. Li, and Y. H. Shi, “Evolutionary computation meets machine learning: A survey,” IEEE Computational Intelligence Magazine, vol. 6, no. 4, pp. 68-75, Nov. 2011. [Related to Chapters 1&6: Survey of EC&ML]


[13]. Zhi-Hui Zhan, X. Liu, H. Zhang, Z. Yu, J. Weng, Y. Li, T. Gu, and J. Zhang, “Cloudde: A heterogeneous differential evolution algorithm and its distributed cloud version,” IEEE Transactions on Parallel and Distributed Systems, DOI: 10.1109/TPDS.2016.2597826, 2016. [Related to Chapter 6: Future work]


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Author’s Key Publications

Monograph

[1]. J. Zhang, Zhi-Hui Zhan, W. N. Chen, J. H. Zhong, N. Chen, Y. J. Gong, R. T. Xu, and Z. Guan, Computation Intelligence, Tsinghua University Press, November, 2011.

[2]. J. Zhang, W. N. Chen, X. M. Hu, Y. Lin, W. L. Zhong, Zhi-Hui Zhan, and T. Huang, Numerical Computing，Tsinghua University Press, July, 2008.

ESI Hot Paper

[1]. Zhi-Hui Zhan, J. Zhang, Y. Li, and Y. H. Shi, “Orthogonal learning particle swarm optimization,” IEEE Transactions on Evolutionary Computation, vol. 15, no. 6, pp. 832-847, Dec. 2011. (IF=5.908; Citation: Google Scholar 335 times, SCI 196 times; The Top 3 cited paper of this journal in recent 5 years, since 2011)


[2]. Zhi-Hui Zhan, J. Zhang, Y. Li, and H. Chung, “Adaptive particle swarm optimization,” IEEE Transactions on Systems, Man, and Cybernetics--Part B, vol. 39, no. 6, pp. 1362-1381, Dec. 2009. (IF=4.943; Citation: Google Scholar 990 times, SCI 502 times; The Top 3 cited paper of this journal in recent 10 years, since 2006)

[3]. Y. H. Li, Zhi-Hui Zhan(Corresponding Author), S. Lin, J. Zhang, and X. N. Luo, “Competitive and cooperative particle swarm optimization with information sharing mechanism for global optimization problems,” Information Sciences, vol. 293, no. 1, pp. 370-382, 2015. (IF=3.364; Citation: Google Scholar 26 times)

[4]. W. Chen, J. Zhang, Y. Lin, N. Chen, Zhi-Hui Zhan, H. Chang, Y. Li, and Y. H. Shi, “Particle swarm optimization with an aging leader and challengers,” IEEE Transactions on Evolutionary Computation, vol. 17, no. 2, pp. 241-258, April. 2013. (IF=5.908; Citation: Google Scholar 130 times, SCI 41 times)

Other Journal Papers

[5]. X. F. Liu, Zhi-Hui Zhan(Corresponding Author), D. Deng, Y. Li, T. L. Gu, and J. Zhang, “An energy efficient ant colony system for virtual machine placement in cloud computing,” IEEE Transactions on Evolutionary Computation, DOI: 10.1109/TEVC.2016.2623803. 2016. (IF=5.908)

[6]. Y. Li, Y. Zhou, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “A primary theoretical study on decomposition-based multiobjective evolutionary algorithms,” IEEE Trans. on Evolutionary Computation, vol. 20, no. 4, pp. 563-576, Aug. 2016. (IF=5.908)

[7]. Q. Lin, J. Chen, Zhi-Hui Zhan, W. Chen, C. Coello Coello, Y. Yin, C. Lim, and J. Zhang, “A hybrid evolutionary immune algorithm for multiobjective optimization problems,” IEEE Transactions on Evolutionary Computation, vol. 20, no. 5, pp. 711-729, Oct. 2016. (IF=5.908)

[8]. X. Zhang, J. Zhang, Y. Gong, Zhi-Hui Zhan, W. Chen, and Y. Li, “Kuhn-munkres parallel genetic algorithm for the set cover problem and its application to large-scale wireless sensor networks,” IEEE Transactions on Evolutionary Computation, vol. 20, no. 5, pp. 695-710, Oct. 2016. (IF=5.908)


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[9]. Y. J. Gong, J. Zhang, H. Chung, W. N. Chen, Zhi-Hui Zhan, Y. Li, and Y. H. Shi, “An efficient resource allocation scheme using particle swarm optimization,” IEEE Transactions on Evolutionary Computation, vol. 16, no. 6, pp. 801-816, Dec. 2012. (IF=5.908; Citation: Google Scholar 34 times, SCI 16 times)

[10]. Zhi-Hui Zhan, J. Li, J. Cao, J. Zhang, H. Chung, and Y. H. Shi, “Multiple populations for multiple objectives: A coevolutionary technique for solving multiobjective optimization problems,” IEEE Transactions on Cybernetics, vol. 43, no. 2, pp. 445-463, April. 2013. (IF=4.943; Citation: Google Scholar 68 times, SCI 32 times)

[11]. Y. L. Li, Zhi-Hui Zhan(Corresponding Author), Y. J. Gong, J. Zhang, Y. Li, and Q. Li, “Fast micro-differential evolution for topological active net optimization,” IEEE Transactions on Cybernetics, vol. 46, no. 6, pp. 1411-1423, Jun. 2016. (IF=4.943; Citation: Google Scholar 3 times)

[12]. Y. L. Li, Zhi-Hui Zhan(Corresponding Author), Y. J. Gong, W. N. Chen, J. Zhang, and Y. Li, “Differential evolution with an evolution path: A DEEP evolutionary algorithm,” IEEE Transactions on Cybernetics, vol. 45, no. 9, pp. 1798-1810, Sept. 2015. (IF=4.943; Citation: Google Scholar 13 times, SCI 1 times)

[13]. N. Chen, W. N. Chen, Y. J. Gong, Zhi-Hui Zhan, J. Zhang, Y. Li, and Y. S. Tan, “An evolutionary algorithm with double-level archives for multiobjective optimization,” IEEE Transactions on Cybernetics, vol. 45, no. 9, pp. 1851-1863, Sept. 2015. (IF=4.943; Citation: Google Scholar 8 times, SCI 1 times)

[14]. W. J. Yu, M. Shen, W. N. Chen, Zhi-Hui Zhan, Y. J. Gong, Y. Lin, O. Liu, and J. Zhang, “Differential evolution with two-level parameter adaptation,” IEEE Transactions on Cybernetics, vol. 44, no. 7, pp. 1080-1099, Jul. 2014. (IF=4.943; Citation: Google Scholar 31 times, SCI 12 times)

[15]. Zhi-Hui Zhan, X. Liu, H. Zhang, Z. Yu, J. Weng, Y. Li, T. Gu, and J. Zhang, “Cloudde: A heterogeneous differential evolution algorithm and its distributed cloud version,” IEEE Transactions on Parallel and Distributed Systems, vol. 28, no. 3, pp. 704-716, March. 2017. (IF=2.661)

[16]. Zhi-Hui Zhan, J. Zhang, Y. Li, O. Liu, S. K. Kwok, W. H. Ip, and O. Kaynak, “An efficient ant colony system based on receding horizon control for the aircraft arrival sequencing and scheduling problem,” IEEE Transactions on Intelligent Transportation Systems, vol. 11, no. 2, pp. 399-412, Jun. 2010. (IF=2.534; Citation: Google Scholar 78 times, SCI 33 times)

[17]. M. Shen, Zhi-Hui Zhan(Corresponding Author), W. N. Chen, Y. J. Gong, J. Zhang, and Y. Li, “Bi-velocity discrete particle swarm optimization and its application to multicast routing problem in communication networks,” IEEE Transactions on Industrial Electronics, vol. 61, no. 12, pp. 7141-7151, Dec. 2014. (IF=6.383; Citation: Google Scholar 33 times, SCI 11 times)

[18]. Y. J. Gong, M. Shen, J. Zhang, O. Kaynak, W. N. Chen, and Zhi-Hui Zhan, “Optimizing RFID network planning by using a particle swarm optimization algorithm with redundant reader elimination,” IEEE Transactions on Industrial Informatics, vol. 8, no. 4, pp. 900-912, Nov. 2012. (IF=4.708; Citation: Google Scholar 47 times, SCI 27 times)

[19]. Zhi-Hui Zhan, X. F. Liu, Y. J. Gong, J. Zhang, H. S. H. Chung, and Y. Li, “Cloud computing resource scheduling and a survey of its evolutionary approaches,” ACM Computing Surveys, vol. 47, no. 4, Article 63, pp. 1-33, Jul. 2015. (IF=5.243; Citation: Google Scholar 10 times, SCI 1 times)


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[20]. Q. Liu, W. Wei, H. Yuan, Zhi-Hui Zhan(Corresponding Author), and Y. Li, “Topology selection for particle swarm optimization,” Information Sciences, vol. 363, no. 1, pp. 154-173, Oct. 2016. (IF=3.364)

[21]. J. Zhang, Zhi-Hui Zhan, Y. Lin, N. Chen, Y. J. Gong, J. H. Zhong, H. S. H. Chung, Y. Li, and Y. H. Shi, “Evolutionary computation meets machine learning: A survey,” IEEE Computational Intelligence Magazine, vol. 6, no. 4, pp. 68-75, Nov. 2011. (IF=3.647; Citation: Google Scholar 77 times, SCI 41 times)

[22]. Y. Gong, W. Chen, Zhi-Hui Zhan, J. Zhang, Y. Li, Q. Zhang, and J. Li, “Distributed evolutionary algorithms and their models: A survey of the state-of-the-art,” Applied Soft Computing, vol. 34, pp. 286-300, Sept. 2015. (IF=2.857; Citation: Google Scholar 10 times, SCI 2 times)

[23]. W. Yu, Zhi-Hui Zhan, and J. Zhang, “Artificial bee colony algorithm with an adaptive greedy position update strategy,” Soft Computing, DOI:10.1007/s00500-016-2334-4. 2016. (IF=1.630)

Selected Conference Papers (First/Corresponding Author, most are CEC/GECCO/SSCI papers)

[1]. Zhi-Hui Zhan, Z. J. Wang, Y. Lin, and J. Zhang, “Adaptive radius species-based particle swarm optimization for multimodal optimization problems,” in Proc. IEEE Congr. Evol. Comput. (CEC 2016), Vancouver, Canada, Jul. 2016, pp. 2043-2048.

[2]. Z. J. Wang, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “Orthogonal learning particle swarm optimization with variable relocation for dynamic optimization,” in Proc. IEEE Congr. Evol. Comput. (CEC 2016), Vancouver, Canada, Jul. 2016, pp. 594-600.

[3]. X. F. Liu, Zhi-Hui Zhan(Corresponding Author), J. H. Lin, and J. Zhang, “Parallel differential evolution on distributed computational resources for power electronic circuit optimization,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2016), 2016, pp. 117-118.

[4]. Z. J. Wang, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “Parallel multi-strategy evolutionary algorithm using massage passing interface for many-objective optimization,” in Proc. IEEE Symposium Series on Computational Intelligence (SSCI 2016), Athens, Greece, Dec. 2016, pp. 1-8.

[5]. Z. G. Chen, Zhi-Hui Zhan(Corresponding Author), W. Shi, W. N. Chen, and J. Zhang, “When neural network computation meets evolutionary computation: A survey,” in Proc. International Symposium on Neural Networks (ISNN 2016), Saint Petersburg, Russia, Jul. 2016, pp. 603-612.

[6]. Y. F. Li, Zhi-Hui Zhan(Corresponding Author), Y. Lin, and J. Zhang, “Comparisons study of APSO OLPSO and CLPSO on CEC2005 and CEC2014 test suits,” in Proc. IEEE Congr. Evol. Comput. (CEC 2015), Sendai, Japan, 2015, pp. 3179-3185.

[7]. Z. G. Chen, K. J. Du, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “Deadline constrained cloud computing resources scheduling for cost optimization based on dynamic objective genetic algorithm,” in Proc. IEEE Congr. Evol. Comput. (CEC 2015), Sendai, Japan, 2015, pp. 708-714.

[8]. H. H. Li, Y. W. Fu, Zhi-Hui Zhan(Corresponding Author), and J. J. Li, “Renumber strategy enhanced particle swarm optimization for cloud computing resource scheduling,” in Proc. IEEE Congr. Evol. Comput. (CEC 2015), Sendai, Japan, 2015, pp. 870-876.

[9]. X. F. Liu, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “Dichotomy guided based parameter adaptation for differential evolution,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2015), Madrid, Spain, Jul. 2015, pp. 289-296.

[10]. H. H. Li, Z. G. Chen, Zhi-Hui Zhan(Corresponding Author), K. J. Du, and J. Zhang, “Renumber coevolutionary multiswarm particle swarm optimization for multi-objective workflow scheduling on cloud computing environment,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2015), Madrid, Spain, Jul. 2015, pp. 1419-1420.


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[11]. Z. J. Wang, Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “An improved method for comprehensive learning particle swarm optimization,” in Proc. IEEE Symposium Series on Computational Intelligence (SSCI 2015), Cape Town, South Africa, Dec. 2015, pp. 218-225.

[12]. Zhi-Hui Zhan and J. Zhang, “Differential evolution for power electronic circuit optimization,” in Proc. Conf. Technologies and Applications of Artificial Intelligence (TAAI 2015), Tainan, Taiwan, Nov. 2015, pp. 158-163.

[13]. Z. G. Chen, Zhi-Hui Zhan(Corresponding Author), H. H. Li, K. J. Du, J. H. Zhong, Y. W. Foo, Y. Li, and J. Zhang, “Deadline constrained cloud computing resources scheduling through an ant colony system approach,” in Proc. Int. Conf. Cloud Computing Research and Innovation (ICCCRI 2015), Singapore, Oct. 2015, pp. 112-119.

[14]. Zhi-Hui Zhan, J. J. Li, and J. Zhang, “Adaptive particle swarm optimization with variable relocation for dynamic optimization problems,” in Proc. IEEE Congr. Evol. Comput. (CEC 2014), Beijing, China, Jul. 2014, pp. 1565-1570.

[15]. X. F. Liu and Zhi-Hui Zhan(Corresponding Author), “Energy aware virtual machine placement scheduling in cloud computing based on ant colony optimization approach,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2014), Vancouver, Canada, Jul., 2014, pp. 41-47.

[16]. G. W Zhang and Zhi-Hui Zhan(Corresponding Author), “A normalization group brain storm optimization for power electronic circuit optimization,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2014), Vancouver, Canada, Jul., 2014, pp. 183-184.

[17]. Zhi-Hui Zhan, G. Y. Zhang, Y. J. Gong, and J. Zhang, “Load balance aware genetic algorithm for task scheduling in cloud computing,” in Proc. Simulated Evolution And Learning (SEAL 2014), Dec. 2014, pp. 644-655.

[18]. Meng-Dan Zhang, Zhi-Hui Zhan(Corresponding Author), J. J. Li, and J. Zhang, “Tournament selection based artificial bee colony algorithm with elitist strategy,” in Proc. Conf. Technologies and Applications of Artificial Intelligence (TAAI 2014), Taiwan, Nov. 2014, pp. 387-396.

[19]. Guang-Wei Zhang, Zhi-Hui Zhan(Corresponding Author), K. J. Du, Y. Lin, W. N. Chen, J. J. Li, and J. Zhang, “Parallel particle swarm optimization using message passing interface,” in Proc. The 18th Asia Pacific Symposium on Intelligent and Evolutionary Systems (IES 2014), Singapore, Nov. 2014, pp. 55-64.

[20]. Y. L. Li, and Zhi-Hui Zhan(Corresponding Author), and J. Zhang, “Differential evolution enhanced with evolution path vector,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2013), Jul., 2013, pp. 123-124.

[21]. Zhi-Hui Zhan, W. N. Chen, Y. Lin, Y. J. Gong, Y. L. Li, and J. Zhang, “Parameter investigation in brain storm optimization,” in Proc. IEEE Symposium Series on Computational Intelligence (SSCI 2013), Singapore, April. 2013, pp. 103-110.

[22]. Zhi-Hui Zhan, J. Zhang, Y. H. Shi, and H. L. Liu, “A modified brain storm optimization,” in Proc. IEEE Congr. Evol. Comput. (CEC 2012), Brisbane, Australia, Jun. 2012, pp. 1-8.

[23]. Zhi-Hui Zhan and J. Zhang “Enhance differential evolution with random walk,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2012), Philadelphia, America, Jul. 2012, pp. 1513-1514.

[24]. Zhi-Hui Zhan, K. J. Du, J. Zhang, and J. Xiao, “Extended binary particle swarm optimization approach for disjoint set covers problem in wireless sensor networks,” in Proc. Conf. Technologies and Applications of Artificial Intelligence (TAAI 2012), Tainan, Taiwan, 2012. pp. 327-331.

[25]. Zhi-Hui Zhan n, and J. Zhang, “Orthogonal learning particle swarm optimization for power electronic circuit optimization with free search range,” in Proc. IEEE Congr. Evol. Comput. (CEC 2011), New Orleans, Jun. 2011, pp. 2563-2570.

[26]. Zhi-Hui Zhan n and J. Zhang, “Co-evolutionary differential evolution with dynamic population size and adaptive migration strategy,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2011), Dublin, Ireland, Jul., 2011, pp. 211-212.


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[27]. Zhi-Hui Zhan and J. Zhang, “Self-adaptive differential evolution based on PSO learning strategy,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2010), Portland, America, Jul., 2010, pp. 39-46.

[28]. Zhi-Hui Zhan and J. Zhang, “A parallel particle swarm optimization approach for multiobjective optimization problems,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2010), Portland, America, Jul., 2010, pp. 81-82.

[29]. Zhi-Hui Zhan, J. Zhang, and Z. Fan, “Solving the optimal coverage problem in wireless sensor networks using evolutionary computation algorithms,” in Proc. Simulated Evolution And Learning (SEAL 2010), LNCS 6457, pp. 166–176, 2010.

[30]. Zhi-Hui Zhan, J. Zhang, and Y. H. Shi, “Experimental study on PSO diversity,” in Proc. 3rd Int. Workshop on Advanced Computational Intelligence (IWACI 2010), Suzhou, China, Aug. 2010, pp. 310-317.

[31]. Zhi-Hui Zhan, X. L. Feng, Y. J. Gong, and J. Zhang, “Solving the flight frequency programming problem with particle swarm optimization,” in Proc. IEEE Congr. Evol. Comput. (CEC 2009), Trondheim, Norway, May. 2009, pp. 1383-1390.

[32]. Zhi-Hui Zhan, J. Zhang, and R. Z. Huang, “Particle swarm optimization with information share mechanism,” in Proc. Genetic Evol. Comput. Conf. (GECCO 2009), Montréal, Canada, Jul., 2009, pp. 1761-1762.

[33]. Zhi-Hui Zhan and J. Zhang, “Parallel particle swarm optimization with adaptive asynchronous migration strategy,” The 9th Int. Conf. on Algorithms and Architectures for Parallel Processing (ICA3PP), Taipei, Taiwan, Jun., 2009, pp. 490-501.

[34]. Zhi-Hui Zhan and J. Zhang, “Discrete particle swarm optimization for multiple destination routing problems,” EvoWorkshops 2009, LNCS 5484, April. 2009, pp. 117–122.

[35]. Zhi-Hui Zhan, J. Xiao, J. Zhang, and W. N. Chen, “Adaptive control of acceleration coefficients for particle swarm optimization based on clustering analysis,” in Proc. IEEE Congr. Evol. Comput. (CEC 2007), Singapore, Sept. 2007, pp. 3276-3282.

Authorized Patents

[1]. J. Zhang, Zhi-Hui Zhan, and T. Huang, Multicast Approach Based on Particle Swarm Optimization, Patent No. ZL200810220650.1

Summary of Key Publications in SCI Journals

Journal Names 5-Years IF IF IF Rank in JCR Category PapersIEEE Trans. Evol. Comput. 6.897 5.908 Computer Science – Theory & Method 1/105 7 IEEE Trans. SMC. Part B (CYB) 4.978 4.943 Computer Science – Cybernetics 1/22 6 IEEE Trans. Ind. Electron. 5.985 6.383 Automation & Control System 1/59 1 IEEE Trans. Intell. Transp.Syst. 3.155 2.534 Transportation Science & Technology 6/33 1 IEEE Trans. Ind. Informatics 4.880 4.708 Automation & Control System 3/59 1 IEEE Trans. Paral. Distr. Syst. 2.749 2.661 Computer Science – Theory & Method 11/105 1 IEEE Comput. Intell. Mag. 3.483 3.647 Computer Science – Artificial Intelligence 18/130 1 ACM Computing Surveys 6.559 5.243 Computer Science – Theory & Method 2/105 1 Information Sciences 3.683 3.364 Computer Science – Information Systems 10/144 2 Applied Soft Computing 3.288 2.857 Computer Science – Interdiscip. Appl. 14/104 1 Soft Computing 1.732 1.630 Computer Science – Artificial Intelligence 57/130 1 Total 23

Abstract

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Abstract

Particle swarm optimization (PSO) is a kind of simple yet powerful optimization technique.

When compared with other evolutionary computation (EC) algorithms such as genetic

algorithm (GA), PSO is simper in the algorithm structure, easier in the implementation, and

faster in convergence. Therefore, PSO has good application prospect in various science and

engineering optimization problems, attracting great interesting and attention from researchers

all over the world. During the almost two decades’ development since PSO was invented in

1995, some key problems as follows emerge and call for urgent solutions.

1) The PSO performance strongly relies on the parameter and operator in different

evolutionary states. How to be aware of the evolutionary states and adaptively control the

parameter and operator to obtain better algorithm performance is a hot yet difficult research

topic in PSO community.

2) Although PSO can obtain a reasonable solution fast for various problems, the fast

convergence speed makes PSO easy to be trapped into local optima, especially in complex

multimodal optimization problem. How to develop a PSO variant with both faster

convergence speed and strong global search ability is a significant yet challenging research

topic in PSO community.

3) When applying PSO to applications such as multi-objective optimization and

engineering optimization problem in practices, how to utilize and combine the problem

characteristics so as to efficiently solve the practical problem is still a challenging problem in

extending PSO to real-world applications.

In response to these issues, this dissertation carries out innovative researches into the

Abstract

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PSO parameter adaptation control, operator orthogonal design, and population

co-evolutionary interaction. In order to make these researches efficient, this dissertation

points out that the characteristics of population-based search and iteration-based evolution in

PSO provide a mass of search data and historical data during the evolutionary process. As

machine learning (ML) technique is a powerful tool for obtaining useful information from

large amounts of data, using ML technique to analyze, process, and ultilize these data has

great significance to aid PSO algorithm design and so as to improve the algorithm

performance. In view of this, this dissertation conducts researches into ML aided PSO and its

engineering application. The main works are to apply the techniques and ideas such as

statistical analysis, orthogonal design and prediction, and ensemble learning in the ML field

to aid PSO design, improving the algorithm performance, and extending its applications.

The main innovative contributions of this thesis are as follows:

(1) Propose a statistical analysis based adaptive PSO (APSO) to make the algorithm can

act properly according to different states, enhancing the algorithm versatility.

The parameter and operator requirements for PSO are different in different evolutionary

states. By using the strong ability of ML technique in obtaining useful information from mass

data, this dissertation proposes to make statistical analyses on the population distribution data

and fitness data of PSO during the evolutionary process. This results in a novel evolutionary

state estimation (ESE) method that can classify different evolutionary states efficiently. By

using the ML technique aided ESE method, APSO can adaptively control the parameters and

operators according to different requirements in different states, improving the PSO

performance and enhancing the PSO versatility in different search environments.

Abstract

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(2) Propose an orthogonal design and prediction based orthogonal learning PSO

(OLPSO) to enhance the algorithm global search ability in complex optimization.

As the learning strategy in traditional PSO can not sufficiently utilize the information in

the personal experience and the neighborhood experience, this dissertation proposes a novel

orthogonal learning (OL) strategy for the particle to construct a promising exemplar to guide

the flying. The OL strategy is based on the orthogonal experimental design technique in ML

that can efficiently discover useful information in the personal and neighborhood experiences

and predict promising combination of these two experiences. Therefore, OLPSO can obtain

both fast convergence speed and strong global search ability. The promising performance of

OLPSO makes it an efficient tool for complex and multimodal optimization problems.

(3) Propose a co-evolutionary multi-swarm PSO (CMPSO) inspired by the ensemble

learning idea in ML, enhancing the performance in multi-objective optimization.

The ensemble learning method in ML is to use multiple classifiers to enhance the

classification ability. Inspired by such multiple learners’ idea, this dissertation designs a novel

optimization framework as multiple populations for multiple objectives (MPMO) when using

EC algorithms to solve multi-objective optimization problems (MOP). Based on the MPMO

framework, the CMPSO algorithm on the one hand avoids the fitness assignment problem

which caused by considering all the objectives together, and on the other hand searches

sufficiently in different areas of the Pareto front (PF) by the guidance of each objective.

Moreover, CMPSO uses a novel external shared archive for the communication and

co-evolution of different swarms, so as to make the non-dominated solutions cover along the

whole PF efficiently, enhancing the performance in MOP.

Abstract

- XVIII -

(4) Apply OLPSO to the power electronic circuits (PEC) design problem, extending the

engineering application fields of PSO.

The PEC design problem is a complex engineering application problem for that it

involves lots of components such as resistors, capacitors, and inductors, which are all needed

optimally designed so as to obtain good circuit performance. This dissertation on the one

hand extends the traditional PEC optimization model by introducing free search range for the

components. Although this new model makes PEC much closer to real-world application, it

brings great challenges to current optimization methods. Therefore, this dissertation on the

other hand proposes to apply the powerful ML aided OLPSO to optimize PEC with free

search range. The successes of OLPSO in the PEC application not only provides PEC a

powerful optimizer, but also demonstrates that ML aided PSO algorithms have great potential

in real-world engineering optimization problems.

In summary, this thesis argues that ML techniques can acquire useful information from

the PSO data, and therefore proposes the statistical analysis, orthogonal design and prediction,

and ensemble learning techniques and/or ideas in the ML field to aid PSO to improve the

convergence speed, solution accuracy, and application extensions. This is also an attempt to

combine the techniques in EC and ML, which are two of the most significant research fields

in computer science.

Key Words： Particle Swarm Optimization, Machine Learning, Adaptation, Orthogonal

Design and Prediction, Multi-objective Optimization, Engineering Application

Content

- XIX -

Contents

Acknowledgments ................................................................................................................................ III Declaration of Thesis Originality ........................................................................................................... V Permission to Quote Copyrighted Material ......................................................................................... VI Brief CV & Publications ..................................................................................................................... VII Abstract ............................................................................................................................................... XV Contents ............................................................................................................................................ XIX List of Figures .................................................................................................................................. XXII List of Tables ................................................................................................................................... XXIII Chapter 1 Introduction ............................................................................................................................ 1

1.1 Motivation ....................................................................................................................... 1 1.2 Overview of PSO ............................................................................................................ 6

1.2.1 Origin and flow structure of PSO ............................................................................ 6 1.2.2 Theory analysis of PSO ........................................................................................... 9 1.2.3 Parameter control of PSO ..................................................................................... 11 1.2.4 Operator design of PSO ........................................................................................ 13 1.2.5 Population interaction of PSO ............................................................................... 15 1.2.6 Discrete optimization of PSO................................................................................ 16 1.2.7 Practical application of PSO ................................................................................. 18

1.3 ML Technique and PSO ................................................................................................ 20 1.3.1 Overview of machine learning .............................................................................. 20 1.3.2 When PSO meets ML ........................................................................................... 21 1.3.3 ML aided PSO ....................................................................................................... 22

1.4 Contributions of the Thesis ........................................................................................... 23 1.5 Organization of the Thesis ............................................................................................ 25

Chapter 2 Adaptive Particle Swarm Optimization Based on Statistic Analysis Techique in Machine Learning ................................................................................................................................................ 29

2.1 Introduction ................................................................................................................... 29 2.2 Evolutionary State Estimation ....................................................................................... 32 2.3 Adaptive Particle Swarm Optimization ........................................................................ 35

2.3.1 Adaptation of the Inertia Weight ........................................................................... 35 2.3.2 Control of the Acceleration Coefficients ............................................................... 36 2.3.3 Elitist Learning Strategy Adaptation ..................................................................... 38

2.4 Benchmark Tests and Comparisons .............................................................................. 39 2.4.1 Benchmark Functions and Algorithm Configuration ............................................ 39 2.4.2 Comparisons on the Solution Accuracy ................................................................ 41 2.4.3 Comparisons on the Convergence Speed .............................................................. 43 2.4.4 Comparisons on the Algorithm Reliability ............................................................ 44 2.4.5 Comparisons Using t-Tests ................................................................................... 46

2.5 Further Analysis of APSO ............................................................................................. 47 2.5.1 Analysis of Parameter Adaptation and Elitist Learning ........................................ 47 2.5.2 Search Behaviors of APSO and Parameter Evolution Analysis ............................ 48 2.5.3 Sensitivity of the Acceleration Rate ...................................................................... 50

Content

- XX -

2.5.4 Sensitivity of the Elitist Learning Rate ................................................................. 51 2.6 Chapter Summary ......................................................................................................... 52

Chapter 3 Orthogonal Learning Particle Swarm Optimization Based on Orthogonal Experiments Design Techique in Machine Learning ................................................................................................. 53

3.1 Introduction ................................................................................................................... 53 3.2 Orthogonal Learning Particle Swarm Optimization ...................................................... 57

3.2.1 Orthogonal Experimental Design .......................................................................... 57 3.2.2 Orthogonal Learning Strategy ............................................................................... 59 3.2.3 Orthogonal Learning Particle Swarm Optimization .............................................. 61

3.3 Experimental Verification and Comparisons ................................................................. 62 3.3.1 Functions Tested .................................................................................................... 62 3.3.2 Compared Algorithm Configuration ..................................................................... 62 3.3.3 Solution Accuracy with Orthogonal Learning Strategy ........................................ 64 3.3.4 Convergence Speed with Orthogonal Learning Strategy ...................................... 67 3.3.5 Comparisons with Other PSOs .............................................................................. 69 3.3.6 Comparisons with Other Evolutionary Algorithms ............................................... 72 3.3.7 Parameter Analysis ................................................................................................ 73 3.3.8 Discussions ........................................................................................................... 74

3.4 Chapter Summary ......................................................................................................... 75 Chapter 4 Multiple Populations for Multiple Objectives: A Co-evolutionary Technique for Solving Multi-objective Optimization Problems based on Ensemble Learning Techique in Machine Learning .............................................................................................................................................................. 77

4.1 Introduction ................................................................................................................... 77 4.2 Multi-objective Optimization Problem ......................................................................... 81

4.2.1 Related concept of MOP ....................................................................................... 81 4.2.2 Related work on MOP ........................................................................................... 81

4.3 CMPSO for MOP .......................................................................................................... 82 4.3.1 CMPSO Evolutionary Process .............................................................................. 82 4.3.2 CMPSO Archive Update ....................................................................................... 83 4.3.3 Complete CMPSO ................................................................................................. 86 4.3.4 Complexity Analysis of CMPSO .......................................................................... 87

4.4 Experimental Veerification and Comparisons ............................................................... 88 4.4.1 Test Problems ........................................................................................................ 88 4.4.2 Performance Metric ............................................................................................... 89 4.4.3 Experimental Settings ........................................................................................... 89 4.4.4 Experimental Results on ZDT Problems ............................................................... 91 4.4.5 Experimental Results on DTLZ and WFG Problems ............................................ 92 4.4.6 Experimental Results on UF Problems ................................................................. 94 4.4.7 The Benefit of Shared Archive .............................................................................. 96 4.4.8 Impacts of Parameter Settings ............................................................................... 99

4.5 Chapter Summary ....................................................................................................... 102 Chapter 5 Orthogonal Learning Particle Swarm Optimization for Power Electronic Circuit Optimization with Free Search Range ................................................................................................ 103

5.1 Introduction ................................................................................................................. 103

Content

- XXI -

5.2 Power Electronic Circuit ............................................................................................. 105 5.3.1 Particle Representation ....................................................................................... 106 5.3.2 Fitness Function .................................................................................................. 106

5.4 Experiments and comparisons .................................................................................... 107 5.4.1 Circuit Configurations ......................................................................................... 107 5.4.2 Algorithm Configurations ................................................................................... 108 5.4.3 Comparisons on Fitness Quality ......................................................................... 109 5.4.4 Comparisons on Optimization Speed and reliability ........................................... 110 5.4.5 Comparisons on Simulation Results ................................................................... 111 5.4.6 Comparisons on Discrete Search Space .............................................................. 114

5.5 Chapter Summary ....................................................................................................... 115 Chapter 6 Conclusion and Future Work .............................................................................................. 117

6.1 Conclusion .................................................................................................................. 117 6.2 Future work ................................................................................................................. 120

6.2.1 More ML techniques and EC algorithms ............................................................ 121 6.2.2 Dynamic ML aided EC algorithm ....................................................................... 121 6.2.3 Distributed ML aided EC algorithm .................................................................... 121 6.2.4 More Engineering Optimizaiton Practice Test .................................................... 122

References ........................................................................................................................................... 123

List of Figure&Table

- XXII -

List of Figures

Fig. 1-1 The journal papers related to PSO and GA in the IEEE Xploer database in recent 10 years..... 4 Fig. 1-2 The papers related to PSO and ACO in the Google Scholar database in recent 10 years.......... 4 Fig. 1-3 The basic flowchart and pseudo-code of PSO algorithm .......................................................... 8 Fig. 1-4 The topology structure of PSO algorithm ............................................................................... 13 Fig. 1-5 The interaction illustration between PSO algorithm and ML technique ................................. 21 Fig. 1-6 The organization structure and relationship illustration .......................................................... 26

Fig. 2-1 The population distributions of PSO during the evolutionary process. ................................... 30 Fig. 2-2 PSO population distribution information quantified by an evolutionary factor f. ................... 32 Fig. 2-3 Fuzzy membership functions for the four evolutionary states. ............................................... 33 Fig. 2-4 The relationship between inertia weight ω and evolutionary factor f. ..................................... 35 Fig. 2-5 The ideal variants of the acceleration coefficients c1 and c2. ................................................... 37 Fig. 2-6 Convergence performance of the 8 different PSOs on the 12 test functions. .......................... 42 Fig. 2-7 Cumulative percentages of the acceptable solutions obtained duiring the evolutionary process.

...................................................................................................................................................... 45 Fig. 2-8 Search behaviors of the APSO on Sphere function: (a) Mean value of ω during the run time

showing an adaptive momentum; (b) Mean values of c1 and c2 adapting to the evolutionary states. ...................................................................................................................................................... 49

Fig. 2-9 Search behaviors of PSOs on Rastrigin’s function: (a) Mean psd during the run time; (b) Plots of convergence during the minimization run; (c) Mean value of the ω during the run time showing an adaptive momentum; (d) Mean values of c1 and c2 adapting to the evolutionary states. ...................................................................................................................................................... 50

Fig. 3-1 The “oscillation” pheronomon caused by traditional PSO learning strategy. .......................... 54 Fig. 3-2 The “two steps forward, one step back” pheronomon caused by traditional PSO learning

strategy. ......................................................................................................................................... 55 Fig. 3-3 The flowchart of OLPSO. ....................................................................................................... 61 Fig. 3-4 Convergence progresses of PSOs with and without OL strategy on unimodal functions. ...... 65 Fig. 3-5 Convergence progresses of PSOs with and without OL strategy on multimodal functions. ... 66 Fig. 3-6 Convergence progresses of PSOs with and without OL strategy on rotated functions. .......... 67 Fig. 3-7 OLPSO performance with different values of G. (a) OLPSO-G. (b) OLPSO-L. .................... 74

Fig. 4-1 Framework of MPMO based algorithm for solving MOP. ...................................................... 79 Fig. 4-2 The archive update process. .................................................................................................... 85 Fig. 4-3 The complete flowchart of CMPSO. ....................................................................................... 86 Fig. 4-4 The final non-dominated solutions of the ZDT problems in all the 30 runs. ........................... 92 Fig. 4-5 The final non-dominated solutions of the WFG problems in all the 30 runs. ......................... 93 Fig. 4-6 The final non-dominated solutions of the UF problems in all the 30 runs. ............................. 95 Fig. 4-7 The final non-dominated solutions found by CMPSO and CMPSO-non-ELS in all the 30 runs.


- XXIII -

...................................................................................................................................................... 97 Fig. 4-8 The mean IGD of CMPSO and CMPSO-non-aBest during the evolutionary process. ........... 98 Fig. 4-9 The final non-dominated solutions found by CMPSO and CMPSO-non-aBest after 1000 FEs.

...................................................................................................................................................... 98 Fig. 4-10 The mean IGD of CMPSO with different population size. ................................................... 99 Fig. 4-11 The final non-dominated solutions found by CMPSO and OMOPSO with different FEs on

ZDT4. .......................................................................................................................................... 100 Fig. 4-12 The mean IGD on DTLZ2 and UF1 of MOPSOs with different ω and different ci. ........... 101

Fig. 5-1 A block diagram of PEC. ....................................................................................................... 105 Fig. 5-2 Circuit schematics of the buck regulator with overcurrent protection. ................................. 108 Fig. 5-3 Mean convergence characteristics of different approaches in optimizing PEC..................... 111 Fig. 5-4 Simulated voltage responses from 0 ms to 90 ms. ................................................................ 112 Fig. 5-5 Simulated current responses from 0 ms to 90 ms. ................................................................. 113

Fig. 6-1 The summury. ........................................................................................................................ 120

List of Tables

Table 1-1 The Researches on PSO Theory ............................................................................................ 10 Table 1-2 The Rank of PSO Papers in Different IEEE Transactions According to SCI Database ........ 19

Table 2-1 The 12 Functions Used in The Comparisons ........................................................................ 39 Table 2-2 The PSO Algorithms Used in the Comparisons .................................................................... 40 Table 2-3 Results Comparisons on Solution Accuracy Among 8 PSOs on 12 Test Functions ............. 41 Table 2-4 Convergence Speed and Algorithm Reliability Comparisons ............................................... 43 Table 2-5 Comparisons Between the APSO and Other PSOs on t-Tests .............................................. 46 Table 2-6 Merits of Parameter Adaptation and Elitist Learning on Search Quality .............................. 47 Table 2-7 Effects of the Acceleration Rate on Global Search Quality .................................................. 51 Table 2-8 Effects of the Elitist Learning Rate on Global Search Quality ............................................. 51

Table 3-1 The Factors and Levels of the Chemical Experiment Example ............................................ 57 Table 3-2 Deciding the Best Combination Levels of the Chemical Experimental Factors Using an

Orthogonal Experimental Design Method .................................................................................... 58 Table 3-3 Sixteen Test Functions Used in the Comparison ................................................................... 63 Table 3-4 PSO Algorithms for Comparison .......................................................................................... 64 Table 3-5 Solutions Accuracy (Mean±Std) Comparisons Between PSOs With and Without OL

Strategy ......................................................................................................................................... 65 Table 3-6 Convergence Speed, Algorithm Reliability, and Success Performance Comparisons. ......... 68 Table 3-7 Search Result Comparisons of PSOs on 16 Global Optimization Functions ........................ 70 Table 3-8 Convergence Speed, Algorithm Reliability, and Success Performance Comparisons Among

Different PSO Variants .................................................................................................................. 71


- XXIV -

Table 3-9 Result Comparisons of OLPSO-L and Some State of the Art Evolutionary Computation Algorithms With The Existing Results Reported in The Corresponding References ................... 73

Table 4-1 Characteristics of the Test Problems ..................................................................................... 88 Table 4-2 Parameters Settings of the Algorithms .................................................................................. 90 Table 4-3 Results Comparisons on the ZDT Problems ......................................................................... 91 Table 4-4 Results Comparisons on the DTLZ and WFG Problems ...................................................... 93 Table 4-5 Results Comparisons on the UF Problems ............................................................................ 94 Table 4-6 Comparisons Between CMPSO and Its Variants CMPSO-non-ELS (CMPSO without ELS

in the Archive Update) and CMPSO-non-aBest (CMPSO without Using Archive Information for Particle Update) ............................................................................................................................ 96

Chapter 1 Introduction

1


1.1 Motivation

Optimization is an appealing topic that has attracted human being since ancient times.

When human being engaging in the productive practice, scientific research, and social

activities, their behaviors are always driven by some specific goals [1][2]. Before the

development of modern mathematics, people mainly search for the optimal solutions to the

problem depend on their experiences. To the 17 century, after Newton inventing calculus,

many real-world problems were modeled as optimization problems (OP). Shown in (1-1) for

a minimization OP, the objective is to find out a feasible solution X in the search space D, so

as to minimize the objective function f:

Min f(X), X∈D (1-1)

The Newton’s calculus provides an effective approach to solve the peak problem such as

(1-1). For example, in the condition of secondary differentiable, we can find the peaks of f by

finding the positions where the gradient is 0 via the Newton’s method. However, the

Newton’s method, and the ones such as the Lagrange multiplier method and the Cauchy’s

gradient-based method, all require the objective function with the good mathematical

characteristics such as differentiable or secondary differentiable, which has greatly limit the

utilization of these methods in practical applications. Moreover, these gradient information

based methods are local optimization algorithms. Therefore, they are promising in unimodal

functions but are difficult to avoid local optima when dealing with multimodal functions.

With the fast developments of modern technology, the optimization problems emerge in

the scientific researches and engineering practices become more and more complex.

Especially with the widely utilization of computers, optimization problems in modern life

turn to multi-variables, multi-modal, multi-constraint, and multi-objective. Taking the

optimally design of power electronics circuit (PEC) for instance, the engineers have to face a

mount of resistors, capacitors, and inductors, which they have to carefully determine the

component values, so as to design an efficient PEC with good performance. Such an


2

optimization problem not only involves multi-variables and multi-modal, but also has its

difficulty that it is hard, if not impossible, to use a objective function to describe the problem.

Therefore, no matter the classic Newton’s method and gradient-based method, or the

traditional optimization methods such as the quadratic programming, the simplex method,

and dynamic programming method, are all deficient when applying to the complex scientific

and engineering optimization problems in modern time. How to solve the global optimization

problems that are lack of good mathematical model and are with complex challenges such as

multi-variables, multi-modal, multi-constraint, and multi-objective, has become a significant

research topic in the optimization field.

Inspired by the biology evolution and intelligent phenomena in nature, the computer

scientists invented a kind of optimization algorithm that emulated the biology evolutionary

mechanisms and social swarm behaviors. Such kinds of population search based and iteration

evolution based optimization algorithms are called evolutionary algorithm (EA). The EA can

be dated from the 1950s when the American scholar Holland proposed the genetic algorithm

(GA) [3][4]. Later in 1960s, American scholar Fogel proposed the evolutionary programming

(EP) [5], and the Germany scholar Rechenberg proposed the evolution strategy (ES) [6]. Due

to the advantages that they did not depend on the mathematical model and characteristics of

the being solved problem, their potential global search ability, and can obtained the optimal

or near-optimal solution in acceptable computational time, these EA approaches have caused

great attention and interests from the researchers all over the world. EA has fast become a

significant research branch in the computer science and artificial intelligence (AI) field [2].

Although EA provides a significant and effective approach to solve the optimization

problems in various scientific and engineering applications, the characteristics that EA is

based on population search and iterative evolution cause a problem that EA is computational

expensive. Until 1980s, computer scientists got inspirations from intelligent phenomena in

the physical annealing process and the human memory, and invented the simulated annealing

(SA) algorithm [7] and tabu search (TS) algorithm [8][9]. As the SA and TS algorithms are

based on single point search, they release the expensive computational burden of EA in a

sense. However, the single point search characteristics make them easy to be trapped into

local optima and therefore are not efficient enough in real-world application. In this sense, the


3

computer scientists still preferred to the population search based optimization approach like

EA. In order to keep the population search characteristics of EA, and together accelerate the

optimization speed, computer scientists invented the swarm intelligence (SI) algorithms

[14][15] such as ant colony optimization (ACO) [10][11] and particle swarm optimization

(PSO) [12][13]. In the literature, EA and SI algorithms are named evolutionary computation

(EC) algorithms [16][17].

The SI algorithms emulate the intelligent behaviors such as ants foraging and birds

schooling in nature. They are a kind of optimization approaches based on the memory search.

The origin of PSO can be dated back to 1995 when Eberhart and Kennedy proposed a kind of

SI algorithm emulating the birds schooling [12][13]. When compared with GA, PSO has

simpler algorithm flowchart, is easier to implement, and can converge to the optimal solution

or near-optimal solution with the help of the guided by the historical search information.

Therefore, PSO can meet the fast optimization speed requirements of real-world applications,

and has been extensively studied since its origin in 1995. Nowadays, PSO has been

successfully applied to various optimization problems in daily life, and show great

development and application potential [18]-[23]. In 2004, Eberhart, who is the inventor of

PSO, together with his colleague Shi organized a special issue on PSO in the IEEE

Transactions on Evolutionary Computation, which is the leading international journal in the

EC field [24]. In that special issue, seven high quality research papers focusing on different

aspects of PSO were published [25]-[31]. It should be noted that this special issue can be

regarded as a milestone in the PSO field because the PSO algorithm has been really widely

accepted by researchers since then. Fig. 1-1 presents the IEEE journal papers with “particle

swarm” or PSO in the title in the IEEE Xploer database. Fig. 1-1 shows that there are only

several journal papers on PSO before 2004, while the paper number rapidly increases after

2004. To the year of 2008, we can see that the PSO paper number is near the GA paper

number. This indicates that more and more researchers have turned their interests from

traditional EC algorithms like GA to the new and efficient PSO algorithm. Especially in the

applications in various engineering optimization problems, more and more publications

report that PSO can obtain better performance than GA does.


4

2003 2004 2005 2006 2007 2008 2009 2010 2011 20120

10

20

30

40

50

60

70

Pape

rs

Year

Particle Swarm Genetic Algorithm

Fig. 1-1 The journal papers related to PSO and GA in the IEEE Xploer database in recent 10 years

The PSO algorithm, together with the SI algorithms, have become one of the hottest

research topics in the EC and computer science fields all over the world. In the international

conferences such as IEEE Congress on Evolutionary Computation（CEC）sponsored by IEEE

and The Genetic and Evolutionary Computation Conference（GECCO）sponsored by ACM,

the theory, design, and application of PSO are always the key directions and topics in the

conference call for papers. Moreover, the emergences of international conferences such as

The International Conference on Ant Colony Optimization and Swarm Intelligence（ANTS）

and The International Conference on Swarm Intelligence（ICSI）which are based on ACO and

PSO also show that PSO have been paid close attentions in recent years. Fig. 1-2 presents the

recent 10 year paper numbers, which are obtained in the Google Scholar database by using

“particle swarm” as the keyword. Similar, the recent 10 year paper numbers on ACO, which

are obtained by using “ant colony” as the keyword, are also presented. From the figure we

can see that in 10 years ago, PSO was not as hot as ACO. However, the paper number on

PSO increases rapidly, and surpasses the paper number on ACO in the year of 2009. In 2012,

the paper number on PSO exceeded 20,000.

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

3,000

6,000

9,000

12,000

15,000

18,000

21,000

24,000

Pape

rs

Year

Particle Swarm Ant Colony

Fig. 1-2 The papers related to PSO and ACO in the Google Scholar database in recent 10 years


5

The growing wealth of scientific research and active academic exchange activities in the

PSO field bring important opportunities and challenges to the algorithm design and practical

application of PSO. Due to that the different components of PSO, such as the parameter,

operator, and population, all have significant influences on the algorithm performance, it is an

open work to the research into PSO parameter control, operator design, and population

interaction.

The researches in this dissertation point out that: when using PSO to solve the global

optimization problems, the population search based and iteration evolution based

characteristics of PSO provide a mass of search data and historical dada during the running

time. How to sufficient utilize these data and discover useful information in these data to help

the parameter control, operator design, and population interaction, is an important and

efficient way to enhance the PSO performance. Machine leaning (ML) technique is a kind of

approach that emulates the human being learning behaviors so as to obtain new knowledge

and skill, and to enhance the performance [32]. As the PSO algorithm can provide mass of

search data and historical dada during the running time, using the ML technique to analyze

and process these data can help to obtain useful information, which can be used to help the

algorithm design and problem solving. Using ML techniques to aid PSO is a significant and

promising research topic.

This thesis conducts the research on ML aided PSO and its application. I will apply the

statistical analyzing technique, orthogonal predicating technique, and ensemble learning

techniques in the ML field to help the parameter control, operator design, and population

interaction of PSO, respectively. At last, I will apply the orthogonal predicating technique

aid PSO algorithm to the PEC design and optimization problem, not only to evaluate the ML

aided PSO algorithm in real-world problem, but also extend the application of PSO in the

engineering field.


6

1.2 Overview of PSO

1.2.1 Origin and flow structure of PSO

Swarm animals like bird, kennel, and fish show high organization and regularity in

natural behaviors such as migration and foraging. Many biologists, zoologists, computer

scientists, and behaviorists, and social psychologists, and other researchers conduct deep

study of such social behaviors, and indicate that the swarm intelligent behaviors is a kind of

optimization behavior drawing on the advantages, avoiding disadvantages, and adapting to

environment [33]. Especially in the application of swarm intelligence, the research of social

psychologist Wilson [34] demonstrate that in theory at least, through the process of swarm

foraging, each individual in the swarm will benefit from the discovery and experience

accumulated by all the individuals in the process. To introduce the idea of self-cognition,

social influence, swarm intelligence in social psychology into swarm behaviors with high

organization and regularity, two researchers, Kennedy, a senior social psychology scholar in

department of labor statistics in United States, and Eberhart, a famous electrical engineer in

Purdue University in United States, collaborated to develop an optimization tool for

engineering practice, and invented PSO finally. Obviously, two scholars gave full play to

their professional knowledge and skill in the design of PSO. On the one hand, PSO integrates

the optimization process based on population in evolutionary computation and swarm

intelligence. On the other hand, the self-cognition and social-influence theory in social

psychology are blended into design of PSO. In 1995, as an important branch of EC, two

papers [12][13] related to PSO were published by Eberhart and Kennedy on international

conference, which represented the birth of PSO.

The basic idea of PSO is to simulate the movement of organisms in a bird flock, and find

the optima or near-optima by population searching and iterative evolution. In a bird flock,

each bird find the position of the food through self-search and social cooperation. We

consider such a scene in [2], a group of disperse birds fly randomly to find food, and they do

not know the position of the food in advance, but are able to know the distance between itself

and the food (by the strength of the smells of the foods or others ways). Thus, each bird will


7

continually record and update the nearest position from food in the fly. Meanwhile, through

information communication, they can compare all the position found by all birds and obtain

the best position in the current entire population. So, each bird owns a guidance direction

during flying and combines self-experience and population-experience to adjust its velocity

and position. Flying closer to the food position step by step, all the birds will gather around

the position of food at last.

In PSO, each bird in the flock is regarded as a “particle” in the algorithm. A group of

particles are initialized randomly as feasible solutions in the problem searching space, and

then perform iterative search to find optima. Just like a bird, each particle has a velocity and a

position, and a fitness value defined by fitness function related to the problem. Through

continued iteration, particles update the velocity and position under the impact of personal

historical best solution and current global best solution, explore and exploit the searching

space and find the global best solutions finally.

Follow the idea represented above, PSO assumes that a population of N particles search

optimum in a D-dimension solution space. Each particle i (1≤i≤N) equipped with a position

vector Xi = [Xi1, Xi2, …, XiD] and a velocity vector Vi = [Vi1, Vi2, …, ViD] to represent the

current state of the particle. Meanwhile, each particle record a vector called personal

historical best position pBesti = [Pi1, Pi2, …, PiD]. That is to say, in the evolution progress,

each particle will record the best position (the position with best fitness value) achieved so far

in vector pBesti, and pBesti will be updated when the particle find a better position. Similarly,

the best position of all the personal historical best position pBesti is recorded as globally best

positon gBest = [G1, G2, …, GD]. It should be note that in this thesis, gBest corresponds to the

best one among all pBesti, and the adoption of symbol gBest is to simplify the expression.

Employing the above vectors expression, the basic flowchart of PSO and pseudo-code are

shown in Fig. 1-3.


8

Fig. 1-3 The basic flowchart and pseudo-code of PSO algorithm

Step 1: Initialization. In the D-dimension searching space, each dimension Xid (1≤i≤N，

1≤d≤D) of each particle i is initialized randomly in the searching range [Xdmin, Xdmax],

and its corresponding velocity Vid are generated randomly in range [–Vdmax, Vdmax],

where Vdmax is the maximum velocity of each particle in dimension d and is set as 20%

of the searching range of that dimension in general [18]. After that, for each particle, the

fitness value is evaluated and pBesti is set as the current position Xi while the globally

best position gBest is set as the best position among all pBesti.

Step 2：In each iteration：

step 2.1）For each particle i, using the pBesti and gBest to update its velocity and

position according to Eq. (1-2) and (1-3):

Vid = ωVid + c1r1d(Pid – Xid) + c2r2d(Gd – Xid) (1-2)

Xid = Xid + Vid (1-3)

step 2.2）Evaluate the new position of particle i; if the fitness value of new position is

better than that of pBesti, then pBesti will be replaced by the new position Xi.; if the

new pBesti is better than gBest, then gBest will be set as pBesti;


9

step 2.3）If all particles are updated, then finish this iteration and go to Step 3);

otherwise, go back to step 2.1) to perform the update process of the next particle.

Step 3：If the terminal condition is not met, then go back to Step 2) to enter the next

iteration; otherwise, output the globally best position gBest as the solution of the problem,

and finish the algorithm.

In Eq. (1-2), ω is inertia weight, c1 and c2 are acceleration coefficients, and r1d and r2d

areteo uniformly distributed random numbers independently generated within [0, 1] for the

dth dimension. Note that, if some components Vid of the updated velocity violate the

boundary constraints of velocity [–Vdmax, Vdmax], then the violating Vid should be revised.

Although different kinds of initialization and revising methods have been proposed for PSO

[35], this thesis follows a simple method which initializes the swarm randomly and sets the

violating components to be the corresponding components of the violated bounds. On the

other hand, some updated position Xid obtained by Eq. (1-3) may violate position constraints

[Xdmin, Xdmax], the typical revising methods are to set the violating components to the be

the corresponding violated boundary or regenerate randomly in the search range [36]. In this

thesis, without special note, a special method is used: once there is a exceed the boundary

constraints, the new position is not evaluated to avoid the spread of the infeasible

information. Due to each particle flies under the guidance of feasible pBesti and gBest, it is

expected to fly back to the search space again.

1.2.2 Theory analysis of PSO

Due to the simple implement but effective performance, PSO is explored and improved

by many researchers, and applied in increasingly wide range of areas. In order to describe

clearly the development history and current research work of PSO, the following subsections

will review the current study from several important aspects focusing on parameter control,

operator design, and population interaction firstly, and then introduce the theory analysis,

discrete optimization, and practice application of the algorithm.

The theory analysis of PSO, is firstly attempted by Ender and Mohan in the early 1999

[37]. Later in 2002, France mathematician Clerc and the inventor of PSO Kennedy focused a


10

deeper description and analysis into the mathematical foundation and convergence

mechanisms of PSO in [38]. In addition, Trelea [39] reported the results about investigation,

research, and analysis of the convergence and stability for PSO, and indicated that PSO will

converge to one position of search space but without any guarantee for converging to the

global optimum position, and in a worse case, the converged position may even not be a local

optimum, and the algorithm is trapped into a stagnation in a current best position. In 2006,

different from the static analysis of PSO before, Li ning et al. [40], Kadirkamanathan et al.

[41] and van den Bergh et al. [42] make deep study into dynamic systematical analysis for

the research and mathematical analysis of PSO. In 2011, Fernandez-Martinez and

Garcia-Gonzalo[43] have analyze the stability of PSO on continuous and discrete PSO two

models. The research works on PSO theory are summarized in Table 1-1.

Table 1-1 The Researches on PSO Theory

Year Authors Characteristic Reference

1999 Ender and Mohan The sine wave characteristics of particles in flight was observed, tracked, and analyzed, and expanded into multidimensional space

[37]

2002 Clerc and Kennedy Design a parameter named compressibility factor and use the parameter to accelerate the convergence of PSO

[38]

2003 Trelea Indicate the steady convergence to a position in the searching space but no guarantee of PSO for converging to the global optimum position

[39]

2006 Li ning et al. Describe the trajectory of the particle and analyze the convergence by differential equations and Z-transform

[40]

2006 Kadirkamanathan et al. Behavior research of PSO in dynamin environment, deep study of static analysis into dynamic analysis

[41]

2006 F. van den Bergh et al. Track the flight trajectory of PSO, and make dynamic systematical analysis and convergence study

[42]

2011 Fernandez-Martinez and

Garcia-Gonzalo

Stability analysis of on continuous and discrete PSO by using stochastic difference and difference equation

[43]


11

1.2.3 Parameter control of PSO

One important reason of the rapid popularity and being widely accepted of PSO is the

fewer number of relative parameters compared with GA or other EC [44]. It can be seen from

Eq. (1-2) that the parameters of PSO mainly includes inertia weight ω and acceleration

coefficients c1 and c2.

In the original PSO, there is no parameter ω in Eq. (1-2). However, in the later research,

Shi and Eberhart [45] discovered that the introduction of inertia weight ω in velocity update

equation of PSO benefits to maintain the continuity of the algorithm searching. They

introduced the parameterω into the velocity update equation of PSO the first time and

indicated that larger value of ω benefits to the search in large range while small value of ω

ensures that the algorithm is able to converge to the optimum position. Therefore, in order to

give the algorithm stronger global searching ability in the early stage and greater local

searching capability, they proposed a model of ω linearly decreasing with iterative

generations as

ω = ωmax – (ωmax – ωmin)t/T (1-4)

where ωmax=0.9 and ωmin=0.4 represent the maximal and minimal inertia weights,

respectively. t and T are the current number of evolutionary generations and maximal number

of generations, respectively. Except the parameter setting method of linearly decreasing, a

fuzzy adaptive [46] and random version setting [47][48] methods are also proposed. To

compare the advatage of different models, Liu et al. analyzed different inertia weight seeting

mothods in [49].

After the introduction of ω by Shi and Eberhart in 1998, Clerc introduced the

constriction factor χ for ensuring the convergence of PSO in 1999 [50], by modifying

velocity update equation to Eq. (1-5a) and Eq. (1-5b). The constriction factor greatly

improves the performance of PSO on unimodal function solving, but not obvious on

multimodal function. According to the research of Eberhart in [51], the constriction factor is

equivalent to the inertia weight mathematically.

Vid = χ[Vid + c1r1d(Pid – Xid) + c2r2d(Gd – Xid)] (1-5a)


12

42

22 ϕϕϕ

χ−−−

= (1-5b)

ϕ = c1 + c2 (1-5c)

Since the inertia weight can dynamicly balance the global and local search ability, and

better performance than constriction factor on multimoal and complex problems, it is more

widely accepted by researchers. The version of PSO adopted to study in this thesis is based

on the version with inertia weight.

In addition to inertia weight and constriction factor, two other important parameters of

PSO are acceleration coefficients c1 and c2. In the early stage of development of PSO, c1 and

c2 are set as 2.0 because the expected value of 2.0 multiplied by a random number in [0, 1] is

1.0, which make the algorithm search solution space steadily. Kennedy [52] investigated the

effect of the two coefficients by designing two models only with self-cognition or only with

social influence acceleration ability, and reported that the reasonable settings of c1 and c2 is an

important issue to ensure the performance of PSO. Suganthan [53] also discovered that

different acceleration coefficients are required for different optimization problems to obtain

optimum solution. Based on these observation, in order to improve the adaptive ability of

PSO on different optimization problems, Ratnaweera et al. [28] proposed a method linearly

increase or decrease: with the iterative generation, c1 linearly decreases while c2 linearly

increases, which is expected to increase the diversity of the swarm by learning more from

personal experience and less global information in the early stage of the algorithm and

accelerate convergence by using more global information in the later stage.

Many researches shows that, in the process of PSO solving optimization problems, and

the different evolution stage in the run of the algorithm, different parameters are suitable.

Although many researchers have proposed a number of setting methods, how to perceive the

evolutionary state of the algorithm and adaptive control the parameters according to the

current state of the algorithm, is still a challenging hot and hard problem. Thus, this thesis

takes this problem as one part of our research.


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1.2.4 Operator design of PSO

The operators of PSO mainly include velocity update and position update operators,

which are simple and represented as Eq. (1-2) and (1-3), respectively. Each particle adjusts its

velocity and position under the guidance of the two update operators as (1-2) and (1-3) and

gets closer to the global best solution. Fig. 1-3 and (1-2) describes the global PSO (GPSO)

which search the optimum solution under the guidance of global best information gBest. Fig.

1-4(a) shows the structure of GPSO. We can that the neighbor of each particles is the entire

population. According to the definition of the neighbor of particle, PSO has corresponding

local version, local PSO (LPSO). A LPSO with ring structure is shown in Fig. 1-4(b), in

which the each particle has two neighbor particles with index of (i – 1) and (i + 1). In another

typical topology structure of LPSO illustrated in Fig. 1-4(c), each particle consider the above,

below, left, and right four neighboring particles as neighbors on the planar mesh. This

structure is known as Von Neumann structure and corresponding to the Von Neumann PSO

(VPSO).

In the local version of PSO, the velocity update operator of particle i is not guided by the

global best solution among the entire population but by the best pBest in the corresponding

neighborhood of particle i (include all neighbors and itself). As opposed to the global best

solution gBest, the guidance vector is named neighborhood best position, and denoted as

nBesti = [Ni1, Ni2, …, NiD] in this thesis. The corresponding velocity update operator is as:

Vid = ωVid + c1r1d(Pid – Xid) + c2r2d(Nid – Xid) (1-6)

Fig. 1-4 The topology structure of PSO algorithm

In the research work related to update operators in PSO, the researchers mainly focus on

designing a reasonable and effective velocity update operator to speed up the process of


14

finding the global best solution under better guidance. In early 2002, one of the PSO

inventors Kennedy published a paper on evolutionary computation international conference

CEC [54] to discuss the performance of different topology structure of PSO algorithm such as

star, ring, castellation, pyramid, and Von Neumann structure. In 2004, Kennedy and

Mendes[55] test these topology structure on fully informed PSO [25]. Much research

discovered that due to the different topology structure impacts the way of information

communication and the speed of information flow, topology structure has a great effect on

performance of PSO. In general, GPSO with star structure has rapid convergence ability and

performs well on unimodal problem but trapped into local optima easily on multimodal

problem. LPSO with ring structure is able to maintain good diversity on complex multimodal

problem and avoid trapped into local optima early due to the slow spread of information. The

structure like VPSO, whose communication density between LPSO and GPSO, obtain a good

balance between rapid convergence and maintaining population diversity.

In addition to obtain different algorithm performance by designing different topology

structure, some researchers have make many other attempts to improve velocity update

operator in PSO. For example, Suganthan [53] proposed a dynamic neighborhood extension

model to select guidance vector for velocity update, in which the neighborhood of each

particle starts with self but grows gradually to the entire population at the end. Hu and

Eberhart [56] defined the particle neighborhood according to physical position and choose the

best particle as the guidance vector from the set of nearest particles evaluated by Euclidean

distance. Liang and Suganthan [57] group the particles randomly as neighborhood, and the

grouping dynamic vary, and the best particle in each group acts as nBest of that group.

Kennedy [58] and Mendes et al. [25] employed the sum of the personal experience of all

particles in neighborhood to drive the fly of the particle. Besides, comprehensive learning

PSO (CLPSO) proposed by Liang et al. in 2006 is a widely accepted improved algorithm, and

consider as one of a typical representative of the improvement of velocity update operator. In

CLPSO, particle not only needs to learn from personal and swarm searching experience but

also from other particles. Therefore, to update the velocity of particle i, CLPSO combines the

pBest of some particles and selects one or more dimensions from each particle to construct a

guidance vector for velocity update. CLPSO demonstrate good performance on multimodal


15

problem. The stochastic combination method increases the population diversity but slows

down the convergence of the algorithm. Therefore, CLPSO performs poorly on unimodal

problem.

Much research works demonstrate that, the operator design of PSO, especially for the

design of velocity update equation, how to effectively use the search information produced in

the evolutionary process to guide the particles fly quickly and accurately is a challenging hot

topic and a hard problem. Thus, an important part of the research work in this thesis, is to

assist the orthogonal design and orthogonal prediction technique in machine learning to

discover the useful information hidden in the found solutions through the evolution process,

and then use the information to construct an efficient velocity update operator to enhance the

rapid global convergence capacity of the algorithm.

1.2.5 Population interaction of PSO

Velocity update or other operator may not be able to make full use of the population

information and cause the stagnation on local optimum position of , another important point

of the easily trapped into local optimum is the lack of the full interaction resulting in the

population premature. To avoid the premature convergence of the population, researchers

proposed different kinds of methods, such as hybrid algorithm and multiple population

algorithm, to strengthen the population interaction.

In hybrid algorithms, the population of PSO interact with other operators or algorithms

to improve the performance. For example, Angeline [60] may be the first one to propose a

method combining PSO with selection operator in GA in 1998. Later, Lovbjerg et al. [61],

Chen et al. [62]and Liang et al. [63] also mixed PSO algorithm with crossover operator.

Hybrid PSO with mutate operator [64][65][66], with local search [67][68], and with double

point expression [69] also have been designed later. In addition, ageing mechanism in nature

[70], chaotic mechanism [71][72][73], quantum mechanism [74][75][76], niching technique

[77][78], speciation pheromone [79] and other methods are used for hybrid to improve the

performance of PSO. Through the interaction with these mechanism, the population

composition and structure of particles will change in the communication process, and this is


16

able to avoid population premature. Furthermore, some researchers proposed hybrid PSO

with other evolutionary computation algorithms, for example, hybrid PSO with simulated

annealing [80][81], with GA [82], with ACO [83][84], with artificial immune algorithm

[85][86], and with differential evolution [87]. The new hybrid algorithm combines the rapid

convergence capacity of PSO with other search feature of other algorithms to enhance the

direct interaction between algorithms and improve the performance.

For multipopulation hybrid algorithms, we can interpret two classes of multipopulation

PSO algorithms as follow. The first class is based on population decomposition: the

population are divided into multiple subpopulation, and each subpopulation employs the

same PSO algorithm to address the same optimization problem [88][89]. Subpopulations can

interact with each other through related communication mechanism or information sharing

technique [90][91]. The second class is based on problem decomposition: the problem is

decomposed into a sum of subproblem, and then different algorithms are used to optimize the

subproblem. Multiple populations requires tight coupling communication and information

sharing mechanism to ensure that the solution information obtained by different populations

in different searching space can be spread rapidly [27].

Much research indicates that, in the population interaction process of PSO, one

significant problem is how to implement the cooperative communication and information

sharing between populations. There have been some related work focusing on this problem,

but how to design a simple but effective method is still a hot topic and hard problem for PSO

[92][93]. Thus, one important part of this thesis is to develop the multiple population

interaction research and design a cooperative communication and information sharing

mechanism to enhance the interaction effect of multiple population and improve the

performance of multipopulation PSO on multiobjective optimization problems.

1.2.6 Discrete optimization of PSO

PSO is proposed originally to solve continuous optimization problem. From (1-2) and

(1-3), we can see that the velocity and position update method based on addition, subtraction,

and multiplication is suitable for continuous optimization but difficult applied on discrete


17

optimization. In order to apply PSO on these discrete optimization problem, researchers work

on designing discrete version of PSO.

The earliest PSO discrete version is binary PSO (BPSO) [94] proposed by the inventors

of PSO Kennedy and Eberhart in 1997. BPSO employs a 0/1 coding mode to encode position

vector. Since BPSO still uses the velocity update equation in (1-2), each dimension of the

obtained velocity is possible to be far away from 0 or 1. To address this problem, BPSO uses

sigmoid function to normalize each dimension of the velocity to range [0, 1] which represents

the probability of valuing 1. Later, some researchers combined PSO with angle modulation

[95] and proposed multi-phase PSO [96] to solve binary optimization problem. Binary PSO

are applied on resource scheduling [97], optimal coverage problem [98] and disjoint set

covers problem [99] in wireless sensor networks, and multiple destination routing problem in

computer network [100].

PSO with Integer coding is also another typical discrete PSO. Salman et al. [101]

adopted a integer encode mode by rounding the real number of position in PSO to the

approximately equal but feasible discrete integer. Yoshida et al. [102] proposed a continuous

space decomposition method, where each area is assigned with a corresponding discrete

integer. These two methods are widely accepted by researchers, and have been successfully

applied to a number of integer discrete optimization problems [103]. The advantages of these

methods lie at no needing of transformation of velocity and position update equation, and

PSO still uses the original method to optimize in the continuous searching space, and just

transforms the continuous position into corresponding discrete value when evaluating fitness

value.

However, the above methods are difficult to solve some discrete combination problem,

so many researchers concern how to modify the velocity and position update equation in

original PSO or redefine the operators designed for continuous space to adapt to operation in

discrete space. Schoofs and Naudts [104] redefined the addition, subtraction, and

multiplication in PSO, and proposed a discrete PSO which successfully solve the constraint

satisfaction problem. Hu et al. [105] defined the velocity as exchange probability of position

variables to solve pawn exchanging in n-queen problem. Clerc [106] also modified and

redefined the operation in velocity and position update equations to implement a discrete


18

PSO to solve the traveling salesman problem.

Chen et al. [107] employed the set-based method to describe the solution spcae in

discrete combinatorial optimization problem, redefined the related operation in velocity and

position update equation in the set spcae, and proposed a novel set-based PSO. The set-based

PSO inherits the learning idea of velocity and position update in continuous space, and

successfully extends the PSO to the discrete space on the condition of retaining the advantage

of rapid convergence rate of original PSO. It demonstrates promising performance on

traveling salesman problem and multidimensional knapsack problem. Gong et al. [108] used

the discrete PSO to optimize the vehicle routing problem and achieved better solutions

compared with the best solution found before on some instances. Zhu and Wang [109] also

proposed a relevance sorting and depth sorting method to transform the solution space of

PSO, and applied it successfully to multiobjective grid scheduling discrete optimization

problem.

Furthmore, combining the characteristics of the solved problem, some researchers

proposed another solution mentality of combining PSO with other methods designed for

discrete optimization problem. Wang et al. [110] developed a discrete PSO based on

estimation of distribution for terminal assigment problems. Tian and Liu [111] employed a

hybrid PSO with iterative greedy algorithm to solve permutation flow shop scheduling

problem. Zhang et al. [112] combined PSO with simulated annealing for shop scheduling

problem. AlRashidi and El-Hawary [113] presented a hybrid PSO with Newton’s method to

address discrete OPF problem. Gao. et al. [114] mixed PSO with genetic operators in GA for

traveling salesman problem. More related work on discrete PSO and its application are

referred to a related review [115].

1.2.7 Practical application of PSO

With the constantly improvement and perfection of PSO, PSO is applied to more and

more area. As a continuous optimization method, PSO mainly is able to solve nearly all the

practical application problems. In many areas where GA has been applied well, PSO can

obtain better solutions and fast convergence speed, decrease the program complexity, and


19

increase the efficiency of algorithm application.

PSO was first used to optimize the connected weights in neural network [116] and

applied to medical diagnosis. The present PSO application cover power system [102],

electromagnetism [118], economic dispatch [119][120], biomedical image registration [31],

system design [121][122], machine learning and training [30][123], data mining and

classification [124][125], pattern recognition [18], signal control [126], flow shop scheduling

[127], mass spectrometers optimization [128]. Table 1-2 The Rank of PSO Papers in Different IEEE Transactions According to SCI Database

Rank Publication Record %, total 217 Histogram 1 IEEE Transactions on Power Systems 35 16.129 % 2 IEEE Transactions on Evolutionary Computation 31 14.286 % 3 IEEE Transactions on Magnetics 24 11.060 % 4 IEEE Transactions on Antennas and Propagation 23 10.599 % 5 IEEE Transactions on Systems Man and Cybernetics B 15 6.912 % 6 IEEE Transactions on Industrial Electronics 12 5.530 % 7 IEEE Transactions on Energy Conversion 7 3.226 % 8 IEEE Transactions on Geoscience and Remote Sensing 7 3.226 % 9 IEEE Transactions on Systems Man and Cybernetics A 7 3.226 %

10 IEEE Transactions on Industrial Informatics 6 2.765 %

Table 1-2 lists the rank of the top 10 IEEE Transactions reported by SCI database

according to the analysis function of the retrieved data under the title keyword of “particle

swarm”, publication keyword of “IEEE Transactions on*”, time interval of “Jan. 1st, 1995” to

“Dec. 31st, 2012”. From the table, we can see that, except the PSO theory study and

improvement research papers on IEEE Transactions on Evolutionary Computation, and IEEE

Transactions on Systems, Man, and Cybernetics, Part B/Part A, the application in other

volumes are mainly related to power system [129], electromagnetic [130], antennas and

propagation [131], industrial electronics [132], energy conversion [121], geoscience and

remote Sensing [133], and industrial informatics [134].

One important part of this thesis is the application development of PSO in power

electronic circuit design. Based on the application features and challenges, using improved

PSO to solve the problem effectively and optimize the engineering application.


20

1.3 ML Technique and PSO

1.3.1 Overview of machine learning

Both machine learning (ML) and evolutionary computation (EC) are important research

area in artificial intelligence (AI) [135]. The origin of ML goes back to the philosophical

issue “Is machine able to think” (another word, can machine learn) mentioned by computer

pioneer Alan Turing in paper [136] in 1950s. During the development of few decades, the

theory and methods have seen substantial development, and become an important tool to

solve information mining and learning problem [137]. The paper “Machine learning for

science: State of the art and future prospects” [138] on Science proposed by scientists at

NASA Jet Propulsion Laboratory in 2001 gave high appreciation and expectation on the

effect of ML in science research. Famous ML sholar Professor. ZhiHua Zhou also made a

detail introduction on current development of ML and data mining in 2007 [139].

Although ML develops rapidly, there is still no uniform strict concepts and definition of

ML. From the intuitive literally, ML is a subject to study how to use machine to simulate

human learning activities. According to the statement of Mitchell, the heart of ML is to get

new knowledge and skill according to the obtained experience to improve the performance

[32]. Thus, ML learns how to achieve new knowledge and new skill and recognite the

existing knowledge. Since experience and information usually is hidden in data, an important

ability of ML is to fetch useful information and knowledge through statistic analysis on the

mass data or other technology, and use them to guide the next work [140].

Based on the strong data process ability and the usful information and knowledge

acquire capability, ML has been widely applied on many areas, such as DNA sequence

sequenced and medical diagnosis, internet search engine design, credit card fraud detection in

economics and finance, security market analysis, speech and handwriting recognition,

strategy games, and robot [141][142]. ML is a promising research topic.


21

1.3.2 When PSO meets ML

ML is an efficient means to guide the performance improvement learning from existing

data. In PSO, due to the algorithm paradigms is based on population searching and iterative

evolution, it produce a large number of searching data and historical data. Tranditional PSO

(or other EC algorithms) usually do not make full use of the data to guide the searching and

running of the algorithm. Indeed, these data is a good souce to get useful information, such as

the searching path of each individual, evolutionary direction of the population, population

distribution, current running state, and the structure characteristics of the found solutions,

interaction in or among population, current advantage of the algorithm, and challenges in the

algorithm. The information, experience, and knowledge are all able to be acquired by

analyzing, predicting, and prcessing the created data in PSO using ML.

Thererfore, as shown in Fig. 1-5, when PSO meets ML, the data created by PSO is a

learning source of ML, and ML offers important assistance for the efficient searching and

running of PSO. ML analyzes and processes the data for PSO, and used it to aid the

implement of the algorithm to improve the performance of PSO.

Fig. 1-5 The interaction illustration between PSO algorithm and ML technique

Therefore, the different features and advantages make the ML and PSO complement one

and another in the study. Since both ML and PSO are two important topic and research area

in AI, the researchers in these two areas focus on their own work to develop algorithm,

technology improvement, and application work. However, in recent years, some researchers

attempted to combine ML and PSO. The works are divided into categories. The first one is to


22

use the optimization ability of PSO to modify the PSO into a more effective ML technology

or to improve the performance of the existing ML techniques. For example, design PSO to be

a efficient game learning tools [30][123], data mining tools [143][144] and classifiers [145].

The second classfier is to introduce the ML into the design of PSO to improve the

performance of PSO. Many researchers do not realize to seek efficient means from ML, but

they use ML techniques unintentionally or deliberately [146]. The following subsection gives

a brief review on related ML aided PSO.

1.3.3 ML aided PSO

When improving PSO, Many researchers have proposed the use of ML techniques such

as statistic analysis, orthogonal experiments analysis, based on inverse learning, and

clustering in PSO.

For the statistic analysis, these research works are mainly based on the data statistic and

analysis technology, and use these techniques to analyze the population position information

[147] and flight velocity data [148] for feeback to control the running of the algorithm.

However, the current findings are not referred to use statistic analysis technique to predict

and evaluate the current state of the algorithms, so the corresponding algorithms lack the

adaptability to different running state.

Researchers employed many different modes using orthogonal experimental design to

improve PSO. Sivanandam and Visalakshi [149] adopted orthogonal experimental design to

improve the initialization of PSO and make initialized solutions uniformly distributed in the

searching space. Ho et al. [150], Liu and Chang [151] combined the personal best and

popualiton best information by orthogonal experimental design to generate a new solution

with better fitness value.

In addition, opposition-based learning is also applied in the initialization of PSO and

solution mutation to produce more new diverse solutions [152][153][154].

Cluster analysis in ML is the most used technology in improving PSO. Janson and

Merkle [155] use cluster analysis to assist PSO in retaining the diversity. They divided the

population into multiple subpopulation and stated that different population need to be a niche


23

to keep independent searching ability. Thus, they adopted clustering in different population to

find and record the corresponding excellent solutions. On the contrary, Pulido and Coello

[156] thinked that it is necessary to enhance the communication between different population

to ensure the performance of the algorithm, so they clustered the optimum solutions in

different population and then used cluster analysis to classfy these optimum solutions into

different classes and assigned the solutions to different population finally. So these population

can learn from the searching information and searching experience of other population.

Kennedy [58] employed cluster analysis technique to decompose the particles into several

classes. Each particle considered the containing class as the neighborhood and used the center

of the class to guide the flight. Similarly, Mei and Zhou [157], and Alizadeh et al. [158] also

suggested that adopting fuzzy clustering to classfy the particles into classes and then using

the center of the class to lead the particle to fly directly. Zhan et al. [159] analyzed the

population distribution data using cluster analysis technique to achieve the adaptive control of

paramters.

It follows that, although some research have use ML to improve PSO, it has not formed

a system of orientation in this area, and it is still an open and chagllenging research topic.

1.4 Contributions of the Thesis

This thesis is to enhance the universality and globality of PSO, and develop research on

parameter adaptive control, operator orthorgonal design, and population collaborative

interactions. The population-based and iterative evolution process features makes PSO create

a greate number of searching data and historical data. Using ML which is able to acquire

useful information and knowledge to analyze, process, and apply these data to give feedback

to the design of the algorithm and the solution space searching to further improve the

algorithm. Based on this method, this thesis tracks the research into ML aided PSO and its

engineering application, and introduce the statistic analysis, orthorgonal prediction, and

ensemble learning into the design of PSO to improve the performance of PSO and extend its

application field.

The main research work and innovation in this thesis are summarized as follow:


24

(1) Based on the statistic analysis technology in ML, this thesis proposes adaptive

PSO (APSO) to improve the universality of the algorithm.

PSO requires different parameters and strategies in different running stage, and makes it

challenging to set the optimum parameters in different evolution stage for different problems.

This thesis uses the capability of of ML techniques discovering useful information in data

statistic analysis to analyze the data pf population distribution and fitness value, and proposes

an evolutionary state estimation (ESE) method to implement the adaptive control of the

algorithm paramters and strategies and improve the universality of PSO in different

optimization problems.

(2) Based on the orthogonal prediction technique, this thesis proposes orthogonal

learning PSO (OLPSO) to enhance the rapid global searching ability of the algorithm.

For the problem that the velocity update equation in tranditional PSO can not make full

use of personal experience and population experience, inspired by the orthogonal design and

orthogonal prediction techniques, this thesis proposes a novel orthogonal learning (OL)

method, to modify the velocity update operator. The OL method discovers usful information

under least computation to predict and construct a guidance vector with optimum searching

experience via orthogonal combination on the personal best experience and population best

experience. OL method further increases the global searching ability of the algorithm, and

makes OLPSO an efficitve and efficient tool to solve large scale and multimodal optimization

problem.

(3) Learning from the idea of ensemble learning in ML, this thesis proposes a

coevolutionary multiswarm PSO (CMPSO) to increase the application effectiveness in

multiobjective problems.

Inspired by the idea of ensemble learning using multiple classifiers to improve the

classifying quality, this thesis introduces the multipopualtion to solve the multiobjective

problmes. Multiple populations for multiple objectives (MPMO) is an optimization

framework which use multiple population to optimize multiple objectives, and each

population solves a fix objective. Based on the MPMO framework, CMPSO escapes from the


25

problem that it is difficult to assign fitness value for each individual in population due to the

multiple objectives. On the other hand, each population evolves under the corresponding

objective, and is able to find good solutions in the corresponding objective space. Meanwhile,

to avoid the unduly impact of the corresponding objective, an information sharing mechanism

is designed to promote the communication between population and collaboration evolution

and make the solutions uniformly distributed on the whole pareto front and increase the

application effect.

(4) Apply the improved OLPSO to power electronic circuit (PEC) design problem

and extend its application in engineering area.

PEC design optimization problem is a complex engineering application problem. There

is a number of resistors, inductors, and capacitors in PEC. How to set the value of the

components is a critical part in circuit design. In traditional ways, the engineers get initial

results by some physical circuit equation operation depending on experience, and then in a

predefined small value range, revise the circuit design step by step through trial-and-error

methods. This method not only needs professional knowledges, but also cause difficulty in

application on complex circuit without exact mathematics model. In order to find an effective

method for PEC design, this thesis studies how to use PSO to optimize the PEC problem, and

proposes a PEC optimization method based on OLPSO. The algorithm not only provides a

new solution but also extends the successful application of PSO in engineering area. This

thesis improve the PEC optimization model, and proposes a “free searching range” model to

address the problem of difficult prediction of the component value range in engineering

practice. Although this model approximates to the real facts, it brings challenges for

optimization algorithms. Therefore, this thesis combines the global earching ability of

OLPSO and the “free searching range” feature in circuit optimization to solve the new PEC

design problem and extend the application of the algorithm in engineering area.

1.5 Organization of the Thesis

There are six chapters in this thesis, and the overall structure is organized as shown in


26

Fig. 1-6.

Fig. 1-6 The organization structure and relationship illustration

The first chapter is the introduction which raises the existing problems in the

development of the algorithms; the second, third, and fourth chapters are about algorithm

research, to solve the problems proposed in Chater 1; the fifth chapter describes the algorithm

application for practice test; and the final chapter is the summary and outlook. The main

contents and organization of each chapter is described as follow:

Chapter 1 is introduction. In this chapter, the original and the flow structure are

introduced firstly, and then the give a survey of the related work about the development

history and current research status of PSO from multiple different views such as theory study,

parameter setting, operator design, population interaction, discrete optimization, and practical

application. To solve the problem existing in parameter control, operator design, and

population interaction, the possibility and advantages of using ML to aid PSO are described.

The current ML aided PSO algorithms are also summarized. After that, the main work and

innovation of the ML aided PSO proposed in this thesis and its application research are

briefly introduced. Finally, the organization of the chapters in the thesis is presented.

Chapter 2 proposes the adaptive PSO based on the statistical analysis in ML. Chapter 2.1

describes the background and motivation of this work, and indicates that the use of the


27

statistical analysis is able to discover information and knowledge and apply from the

population distribution data and fitness values created in the running process of the algorithm.

The information can be used to design an efficient evolutionary state estimation method and

effective adaptive control mechanism for parameters and strategies to improve the

universality of the algorithm in different evolutionary states and different optimization

problems. Chapter 2.2 gives a detail introduction of the evolutionary state based on statistical

analysis. In Chapter 2.3, the adaptive PSO based on evolutionary state estimation is presented

in detail. Chapter 2.4 reports the experiment verification and comparison. In Chapter 2.5, a

further analysis of the search behavior in adaptive PSO is described. Chapter 2.6 is the

chapter summary.

Chapter 3 proposes orthogonal learning PSO based on orthogonal design and prediction

techniques in ML. Chapter 3.1 introduces the background and motivation of this chapter, and

indicates that the traditional learning methods in PSO do not make full the personal and

population searching information, and analyzes the advantages of using the orthogonal design

and prediction technique to discover and utilize personal and population experience, and

predict and construct learning vector to guide the particles converge to global optimum

rapidly, and improve the global searching performance on complex multimodal problems.

Chapter 3.2, gave a detail introduction for the orthogonal learning PSO. Chapter 3.3 is the

experimental verification and comparison. Chapter 3.4 is the chapter summary.

Chapter 4 proposes a co-evolution multipopulation multiobjective (MPMO) PSO

algorithm. Chapter 4.1 firstly describes the background and motivation, and indicates the

possibility and advantage of adopting multiple population to optimize multiple objectives

where one population corresponding to one objective. MPMO not only is able to avoid the

fitness value assignment problem because each individual evolves for all objectives in

traditional multiobjective algorithms, but also make full searching in each objective space. It

benefits to obtain a uniform distribution of solutions along Pareto front and effective

application of PSO in multiobjective problmes. Chapter 4.2 briefly introduces the related

concepts and work of the multiobjective problem. Chapter 4.3 gives a detail description about

the co-evolution multipopulaiton multiobjective PSO algorithm. Chapter 4.4 performs

experiments and comparison. Chapter 4.5 presents a chapter summary.


28

Chapter 5 applies the OLPSO proposed in chapter three to PEC design. Chapter 5.1

firstly describes the background and motivation, and indicates that the PEC optimization

model do not meet the condition that the range of the industrial circuit components is

unpredictable. A “free searching range” PEC optimization model is proposed and the rapid

searching ability of PSO benefits to its application on the new PEC model, and expand the

field of engineering application of PSO and provide a more effective and efficient technique.

Chapter 5.2 introduces briefly the related knowledge of PEC. Chapter 5.3 gives a detail

description of PEC design method based on OLPSO. Experiments and comparison are carried

out in Chapter 5.4. Finally, Chapter 5.5 makes a conclusion.

Chapter 6 summarizes the research work of the whole thesis and give an outlook of the

future work.

Chapter 2 Adaptive Particle Swarm Optimization Based on Statistic Analysis Techique in Machine Learning

29

Chapter 2 Adaptive Particle Swarm Optimization Based on

Statistic Analysis Techique in Machine Learning

2.1 Introduction

Particle swarm optimization (PSO) has been an optimization algorithm widely accepted

by science and engineering application researchers during the development for less than 20

yeas since it is invented. As PSO is easy to implement, it has progressed rapidly in recent

years, and with many successful applications seen in solving real-world optimization

problems. Similar to other EAs, PSO is a population-based iterative algorithm. The simple

concept makes PSO converge more rapidly compared with other EAs such as GA. However,

how to improve the convergence rate of PSO in practical application and avoid trapped in

local optima on complex multimodal problems are still two hot and challenging problems.

Therefore, accelerating convergence speed and avoiding local optima have become the

two most important and appealing goals in PSO research. A number of variant PSO

algorithms have hence been proposed to achieve these two goals. As mentioned in Section

1.2.3, typical methods to control parameters include inertia weight ω linearly decreasing with

the iterative generations introduced by Shi and Eberhart [45] and linearly time-varying

acceleration coefficients method proposed by Ratnaweera [28]. However, since PSO is an

optimization process with guidance and randomness, these linear variety methods fail to meet

the non-linear evolutionary process and have great limitation on application. How to control

both the inertia weight and acceleration coefficients is crucial to improve the performance of

PSO. On the other hand, another active research trend in PSO is hybrid PSO, which combines

PSO with other evolutionary operation, such as selection [60], mutation [64], local search

[67], restart [160], and re-initialization [161]. These hybrid operations are usually

implemented in every generation or are controlled by adaptive strategies using stagnated

generations as a trigger. While these methods have brought improvements in PSO, the

performance may be further enhanced if the auxiliary operations are adaptively performed

with a systematic treatment according to the evolutionary state. For example, the mutation,

reset, and re-initialization operations can be more pertinent when the algorithm has

converged to a local optimum rather than when it is exploring.


30

Thus, to accelerate convergence speed and avoid local optima by parameters control and

combinations with auxiliary techniques, one important problem is how to estimate the

evolutionary state and adaptively control the parameters according to the different

evolutionary states and execution strategies, which is a critical part in the systematic adaptive

control design of the PSO algorithm.

PSO is a population-based iterative optimization algorithm. A great amount of useful

information are offered in the evolutionary process. During a PSO process, the population

distribution characteristics not only vary with the generation number but also with the

evolutionary state. To illustrate the dynamics of the particle distribution in the PSO process,

herein we take a time-varying 2-D Sphere function 2 2

1 2( ) ( ) ( ) , [ 10, 10], 1,2if X r X r X i− = − + − ∈ − =X R (2-1)

as an example, where r is initialized to -5 and shifts to 5 at the 50th generation in a 100

generation optimization process. That is, the theoretical minimum of f1 shifts from (–5, –5) to

(5, 5) half way in the search process. Using a GPSO [45] with 100 particles to solve this

minimization problem, the population distributions in various running phases were observed

as shown in Fig. 2-1.

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10 Particle gBest

X 2

X1

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10 Particle gBest

X 2

X1

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10 Particle gBest

X 2

X1 (a) Generation = 1 (b) Generation = 25 (c) Generation = 49

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10 Particle gBest

X 2

X1

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Particle gBest

X 2

X1

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Particle gBest

X 2

X1 (d) Generation = 50 (e) Generation = 60 (f) Generation = 80

Fig. 2-1 The population distributions of PSO during the evolutionary process.

It can be seen in Fig. 2-1(a) that following the initialization, the particles start to explore

throughout the search space without an evident control center. Then the learning mechanisms

of the PSO pull many particles to swarm together towards the optimal region, as seen in Fig.


31

2-1(b). Then the population converges to the best particle, in Fig. 2-1(c). At the 50th

generation, the bottom of the sphere is shifted from (–5, –5) to (5, 5). It is seen in Fig. 2-1(d)

that a new leader quickly emerges somewhat far away from the current clustering swarm. It

leads the swarm to jump out of the previous optimal region to the new one (Fig. 2-1(e)),

forming a second convergence (Fig. 2-1(f)). From this simple investigation, it can be seen

that the population distribution information can vary significantly during the run time and

that the PSO has an ability to adapt to a time-varying environment. However, the ability of

jumping out the local optima should be enhanced when the algorithm is trapped local optima

and the convergence rate to the new global optimal solution should be improve through

parameter control methods.

Therefore, PSO offeres the population distribution information and fitness data during

the running process. In order to discover useful information and obtain more knowledge from

these data to improve the performance of the algorithm, this chapter used statistical analysis

techniques in machine learning to analyze the search data and historical data including

population distribution and fitness values, and proposed an evolutionary state estimation

(ESE) based on statistical analysis technique to determine the states, and adaptive control the

parameters and execution strategies to accelerate the convergence rate and enhance the global

search ability.

With the help of statistical analysis technique in machine learning, this chapter proposed

an adaptive particle swarm optimization (APSO). The important constributions and

innovations mainly include three aspects as follow:

1) Introduce the statistical analysis technique into PSO algorithm, and design a novel

evolutionary state estimation (ESE) approach based on population distribution information

and fitness data to provide an effective method for adaptive control.

2) Propose an adaptive parameters control strategy of PSO based on ESE approach. The

ESE method is benefit to adaptively control the parameters according to the evolutionary

state, as well as balance the local exploitation and global exploration ability, and accelate the

convergence rate of the algorithm.

3) Design an elitist learning strategy (ELS). ELS is adaptively carried out under the

control of ESE, and increase the population diversity to avoid being trapped in local optima

when the search is identified to be in a convergence state.


32

2.2 Evolutionary State Estimation

Based on the search behaviors and population distribution characteristics of the PSO, an

ESE approach is developed in this subsection. We define an evolutionary factor to describe

the population state. The distribution information in Fig. 2-1 can be formulated as

evolutionary factor illustrated in Fig. 2-2 by calculating the mean distance of each particle to

all the other particles. It is reasonable to expect that the mean distance from the globally best

particle to other particles would be minimal in the convergence state since the global best

tends to be surrounded by the swarm. In contrast, this mean distance would be maximal in the

jumping-out state because the global best is likely to be away from the crowding swarm.

Therefore, the ESE approach will take into account the population distribution information in

every generation, as detailed in the following steps.

ipg dd ≈ipg dd <<

ipg dd >>

5/)(5/)(5141312111

54321

ppppppppgpp

gpgpgpgpgpgdddddd

dddddd++++=

++++=minmax

min

dddd

f g

−−

=

Fig. 2-2 PSO population distribution information quantified by an evolutionary factor f.

Step 1: At a current position, calculate the mean distance of each particle i to all the other

particles. For example, this mean distance can be measured using an Euclidian metric

21

1,

1 ( )1

ND k k

i i jkj j i

d x xN =

= ≠

= −− ∑ ∑ (2-2)

where N and D are the population size and the number of dimensions, respectively

Step 2: Denote di of the globally best particle as dg. Compare all di’s and determine the

maximum and minimum distances dmax and dmin. Compute an ‘evolutionary factor’ f as

defined by

min

max min

gd df

d d−

=−

∈ [0, 1] (2-3)

Step 3: Classify f into one of the four sets S1, S2, S3 and S4, representing the states of


33

exploration, exploitation, convergence and jumping out, respectively. These sets can be

simple crisp intervals, for a rigid classification. From Fig. 2-1 and Fig. 2-2, evolutionary

factor f varies with the evolutionary state, and is able to present the current evolutionary

state. The status transition can be in order during the PSO process, for example, it can

enter exploitation state from exploration state, and then get into convergence state, and

becomes jumping out state when meets local optima, and then enter another loop of

exploration, exploitation, and convergence process. However, the state transition would

be nondeterministic and fuzzy and that different algorithms or applications could exhibit

different characters of the transition. From Fig. 2-1 and Fig. 2-2, we can see that the large

f may indicate the exploration or jumping out states, while the small f value may indicate

the exploitation or convergence states. It is hence recommended that fuzzy classification

be adopted. Combine the advantages of evolutionary factor representation and fuzzy

control, this chapter proposed an evolutionary state estimation method based on fuzzy

classification to assign the running process to one of the four evolutionary states

according to the fuzzy membership functions depicted in Fig. 2-3. Formulation for

numerical implementation of the classification is as follows.

Fig. 2-3 Fuzzy membership functions for the four evolutionary states.

a) Exploration: A medium to large value of f represents S1, whose membership

function is defined as:

1

0, 0 0.45 2, 0.4 0.6

( ) 1, 0.6 0.710 8, 0.7 0.8

0, 0.8 1

S

ff f

f ff f

f

μ

≤ ≤⎧⎪ × − < ≤⎪⎪= < ≤⎨⎪− × + < ≤⎪

< ≤⎪⎩

(2-4a)

b) Exploitation: A shrunk value of f represents S2, whose membership function is defined

as:


34

2

0, 0 0.210 2, 0.2 0.3

( ) 1, 0.3 0.45 3, 0.4 0.6

0, 0.6 1

S

ff f

f ff f

f

μ

≤ ≤⎧⎪ × − < ≤⎪⎪= < ≤⎨⎪− × + < ≤⎪

< ≤⎪⎩

(2-4b)

c) Convergence: A minimal value of f represents S3, whose membership function is

defined as:

3

1, 0 0.1( ) 5 1.5, 0.1 0.3

0, 0.3 1S

ff f f

fμ

≤ ≤⎧⎪= − × + < ≤⎨⎪ < ≤⎩

(2-4c)

d) Jumping out: When PSO is jumping out of a local optimum, the globally best particle

is distinctively away from the swarming cluster, as shown in Fig. 2-2(c). Hence, the

largest values of f reflect S4, whose membership function is thus defined as:

4

0, 0 0.7( ) 5 3.5, 0.7 0.9

1, 0.9 1S

ff f f

fμ

≤ ≤⎧⎪= × − < ≤⎨⎪ < ≤⎩

(2-4d)

The state of the PSO is initialized as Exploration state S1. In each generation, calculate

the value of the evolutionary factor f, and then classify the evolutionary state according to the

following three control rules:

Unique: If f has a degree of only one membership function, then classify the evolutionary

state to the corresponding state. For example, classify f to S3 when f=0.1, and classify f to

S1 when f=0.65.

Stability and proximity: If f has two degree of two membership function, then we will

follow the stability rule first and then the proximity rule to classify the evolutionary state.

For example, an f evaluated to 0.45 has both a degree of membership for S1 and another

degree of membership for S2, indicating that the PSO is in a transitional period between S1

and S2. According the stability rule, we look at the previous state firstly. If the previous

state is S1 or S2, then the algorithm will retain the previous state.

If either the previous state is S1 or S2, then we follow the proximity rule by the sequence

S1 ⇒ S2 ⇒ S3 ⇒ S4 ⇒ S1…. to classify the evolutionary state. if the previous state is S3,

then f is classified to S2. If the previous state is S4, then f is classified to S1. This way, the

algorithm can avoid stochastic oscillator and ensure the coherence of the evolutionary

status transition.


35

2.3 Adaptive Particle Swarm Optimization

2.3.1 Adaptation of the Inertia Weight

The inertia weight ω in PSO is used to balance the global and local search capabilities.

Many researchers have advocated that the value of ω should be large in the exploration state

and be small in the exploitation state. However, the state transition is nonlinear,

nondeterministic, and fuzzy in PSO, traditional parameter control methods based on lineary

change is difficult to ensure the good performance of the algorithm.

From the definition of the evolutionary factor f, we can see that the changing process of f

can describe the running state of the algorithm. In addition, the evolutionary factor f shares

some characteristics with the inertia weight ω in that f is also relatively large during the

exploration state and becomes relatively small in the convergence state. Hence it would be

beneficial to allow ω to follow the evolutionary states. Based on this rule, this chapter designs

an adaptivie transform function using a sigmoid mapping

[ ] ]1,0[,9.0,4.05.111)( 6.2 ∈∀∈

+= − f

ef fω (2-5)

In this chapter, ωis initialized to 0.9. As ω is not necessarily monotonic to time, but

monotonic to f, ω will thus adapt to the search environment characterized by f. In a

jumping-out or exploration state, the large f and ω will benefit global search as referenced

earlier. Conversely, when f is small, an exploitation or convergence state is detected, and

hence ωdecreases to benefit local search. The relationship between ω and f is illustrated in

Fig. 2-4.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.4

0.5

0.6

0.7

0.8

0.9

Iner

tia w

eigh

t w

Evolutionary factor f Fig. 2-4 The relationship between inertia weight ω and evolutionary factor f.


36

2.3.2 Control of the Acceleration Coefficients

According to the research work on acceleration coefficients of PSO, parameter c1 and c2

play different effect on the performance of the algorithm. Parameter c1 represents

‘self-cognition’ that pulls the particle to its own historical best position, helping explore local

niches and maintaining the diversity of the swarm. Parameter c2 represents ‘social-influence’

that pushes the swarm to converge to the current globally best region, helping with fast

convergence. These are two different learning mechanisms and should be given different

treatments in different evolutionary states. In this chapter, the acceleration coefficients are

both initialized to 2.0 and adaptively controlled according to the evolutionary state, with

strategies developed as follow.

Strategy 1: Increasing c1 and decreasing c2 in an exploration state. It is important to

explore as many optima as possible in the exploration state. Hence, increasing c1 and

decreasing c2 can help particles explore individually and achieve their own historical best

positions, rather than crowd around the current best particle that is likely to be associated

with a local optimum.

Strategy 2: Increasing c1 slightly and decreasing c2 slightly in an exploitation state. In

this state, the particles are making use of local information and grouping towards

possible local optimal niches indicated by the historical best position of each particle.

Hence, increasing c1 slowly and maintaining a relatively large value can emphasize the

search and exploitation around pBesti. In the mean time, the globally best particle does

not always locate the global optimal region at this stage yet. Therefore, decreasing c2

slowly and maintaining a small value can avoid the deception of a local optimum.

Further, an exploitation state is more likely to occur after an exploration state and before

a convergence state. Hence changing directions for c1 and c2 should be slightly altered

from the exploration state to the convergence state.

Strategy 3: Increasing c1 slightly and increasing c2 slightly in a convergence state. In the

convergence state, the swarm seems to find the globally optimal region and hence the

influence of c2 should be emphasized to lead other particles to the probable globally

optimal region. Thus the value of c2 should be increased. On the other hand, the value of

c1 should be decreased to let the swarm converge fast. However, such a strategy would

prematurely saturate the two parameters to their lower and upper bounds, respectively.

The consequence is that the swarm will be strongly attracted by the current best region,

causing premature convergence, which is harmful if the current best region is a local


37

optimum. In order to avoid this, both c1 and c2 are increased slightly.

Note that, slightly increasing both acceleration parameters will eventually have the

same desired effect as reducing c1 and increasing c2, because their values will be drawn

to around 2.0 due to an upper bound of 4.0 for the sum of c1 and c2 (refer to Eq. (2-7)

discussed in the following subsection).

Strategy 4: Decreasing c1 and increasing c2 in a jumping-out state. When the globally

best particle is jumping out of local optimum towards a better one, it is likely to be far

away from the crowding cluster. As soon as this new region is found by a particle, which

becomes the (possibly new) leader, others should follow it and fly to this new region as

fast as possible. A large c2 together with a relatively small c1 helps to obtain this goal.

It should be note that the above adjustments on the acceleration coefficients should not

be too irruptive. Hence, the maximum increment or decrement of ci (i=1, 2) between two

generations is bounded by

2,1,)()1( =≤−+ igcgc ii δ (2-6)

where δ is termed the ‘acceleration rate’ in this chapter. δ is a uniformly generated random

value in the interval [0.05, 0.1] and will be regenerated each time used. Note that we use 0.5δ

in strategies 2 and 3 where ‘slight’ changes are recommended.

Further, the interval [1.5, 2.5] is chosen to clamp both c1 and c2 according to the research

work in [38][159]. If the value obtained by Eq. (2-6) violates the bound constraint, then it

will be set as the corresponding violated bound value. Here the interval [3.0, 4.0] is used to

bound the sum of the two parameters. If the sum is larger than 4.0, both c1 and c2 are

normalized to

2,1,0.421

=+

= icc

cc ii (2-7)

The entire process of the ESE enabled adaptive parameter control is illustrated in Fig.

2-5.

Fig. 2-5 The ideal variants of the acceleration coefficients c1 and c2.


38

2.3.3 Elitist Learning Strategy Adaptation

The parameter adaptation of PSO analyzes information generated in the running process

with statistical analysis technique in machine learning, and then learn to adopt suitable and

efficient strategy for parameter adaptive control. However, using parameter adaptation alone

may cause misleading the algorithm towards local optima on multimodal optimization

problem. Hence, an ‘elitist learning strategy’ is designed here and applied to the globally best

particle so as to help jump out of local optimal regions when the search is identified to be in a

convergence state.

The reason of PSO trapped in local optima is due to the lcak of the guidance vector for

the global best solution. With the guidance of the global best solution, the whole population

will converge to gBest, and stagnate at the current best region. Unlike the other particles, the

global leader has no exemplars to follow. It needs fresh momentum to improve itself. Hence,

a perturbation based ELS is developed to help gBest push itself out to a potentially better

region. If another better region is found, the rest of the swarm will follow the leader to jump

out and converge to the new region.

The process of ELS is described in detail as follow:

Step 1: randomly chooses one dimension of gBest’s historical best position, denoted by

Gd for the dth dimension. Only one dimension is chosen because the local optimum is

likely to have some good structure of the global optimum and hence this should be

protected. As every dimension has the same probability to be chosen, the ELS operation

can be regarded to perform on every dimension in a statistical sense. Similar to simulated

annealing, the mutation operation in evolutionary programming or in evolution strategies,

the elitist learning is performed through a Gaussian perturbation

( ) 2max min ( , )d d d dG G X X Gaussian μ σ= + − ⋅ (2-8)

The search range [dX min ,

dX max ] is the same as the lower and upper bounds of the

problem. The Gaussian(μ, σ2) is a random number of a Gaussian distribution with a zero

mean and a standard deviation (SD) σ, termed the ‘elitist learning rate’. Similar to some

time-varying neural network training schemes, it is suggested that σ be decreased linearly

with the generation number, given by

( )max max min /t Tσ σ σ σ= − − ⋅ (2-9)

where σmax =1.0 and σmin =0.1 are the upper and lower bounds of σ, representing the learning

scale to reach a new region.


39

Step 2: Determine whether the obtained Gd is in the range of [Xdmin, Xdmax]. If the value

violates the bound constraint, then Gd is set as the corresponding violated bound value.

Step 3: Evaluate the new gBest. In ELS, the new position will be accepted if and only if

its fitness is better than the current gBest. Otherwise, the new position is used to replace

the particle with the worst fitness in the swarm.

It should be note that, ELS is not performed in each generation, and only carried out

when the algorithm is in a convergence state. This is because that in other evolutionary state,

the algorithm has global search ability and can avoid local optima, and there is no need to

perform ELS. Only when the algorithm is identified to be in a convergence state, no vector

can guide gBest to jump out the local optima, ELS is applied to the globally best particle so

as to help jump out of local optimal regions. Therefore, the execution of ELS is an adaptive

control process.

2.4 Benchmark Tests and Comparisons

2.4.1 Benchmark Functions and Algorithm Configuration

Twelve benchmark functions listed in Table 2-1 are used for the experimental tests here.

Seven existing PSO algorithms, as detailed in Table 2-2, are compared with the APSO. The

first 6 functions f1-f6 are unimodal functions, while the rest 6 functions f7-f12 are multimodal

functions [162]. The dimension of all functions is 30, and the global best fitness value of all

functions are 0 except for Schwefel function f7 is -12569.5. The “Accept” in Table 2-1

indicates whether a solution found by an algorithm falls between the acceptable value and the

actual global optimum. Since not all algorithms can find the global optimum, “Accept” can

be used to measure the successful rate under the predefined error and the corresponding

convergence rate of the algorithm. Table 2-1 The 12 Functions Used in The Comparisons

Function Search range Accept Name

Uni

mod

al

∑ == D

i ixxf1

21 )( [-100,100]D 0.01 Sphere [162]

∑ ∏= =+= D

i

D

i ii xxxf1 12 )( [-10,10]D 0.01 Schwefel’s P2.22 [162]

∑ ∑= == D

i

i

j jxxf1

213 )()( [-100,100]D 100 Quadric [162]

∑ −

= + −+−= 1

1222

14 ])1()(100[)( D

i iii xxxxf [-10,10]D 100 Rosenbrock [162]

⎣ ⎦∑ =+= D

i ixxf1

25 )5.0()( [-100,100]D 0 Step [162]

)1,0[)(1

46 randomixxf D

i i +=∑ = [-1.28,1.28]D 0.01 Quadric Noise [162]


40

Mul

timod

al

∑ = −= Di ixixxf 1 )sin()(7 [-500,500]D -10000 Schwefel [162]

∑ =+−= D

i ii xxxf1

28 ]10)2cos(10[)( π [-5.12,5.12]D 50 Rastrigin [162]

∑ =+−= D

i ii yyxf1

29 ]10)2cos(10[)( π

⎪⎩

⎪⎨⎧

≥

<=

5.0 2

)2(5.0

wherei

i

ii

i xxround

xxy [-5.12,5.12]D 50 Noncontinuous

Rastrigin [59]

exD

xDxfD

i i

D

i i

++−

−−=

∑∑

=

=

20)2cos/1exp(

)/12.0exp(20)(

1

12

10

π

[-32,32]D 0.01 Ackley [162]

∑ ∏= =+−= D

i

D

i ii ixxxf1 1

211 1)/cos(4000/1)( [-600,600]D 0.01 Griewank [162]

∑

∑

=

+−

=

+−+

+−+=

D

i iD

iD

i i

xuy

yyyD

xf

12

1221

112

12

)4,100,10,(})1(

)](sin101[)1()(sin10{)( πππ

⎪⎩

⎪⎨

⎧

−<−−

≤≤−>−

=++=

axaxk

axaaxaxk

mkaxuxy

im

i

i

im

i

iii

,)(

,0 ,)(

),,,( ),1(411 where

[-50,50]D 0.01 Generalized

Penalized [162]

Table 2-2 The PSO Algorithms Used in the Comparisons Algorithm Topology Parameters Settings Reference

GPSO Global Star ω: 0.9-0.4, c1= c2=2.0 [45] LPSO Local Ring ω: 0.9-0.4, c1= c2=2.0 [54]

VPSO Local von Neumann ω: 0.9-0.4, c1= c2=2.0 [54] FIPS Local URing χ=0.729, ∑ci = 4.1 [25]

HPSO-TVAC Global Star ω: 0.9-0.4, c1: 2.5-0.5, c2: 0.5-2.5 [28] DMS-PSO Dynamic Multi-swarm ω: 0.9-0.2, c1= c2=2.0, m=3, R=5 [57]

CLPSO Comprehensive Learning ω: 0.9-0.4, c =1.49445, m=7 [59] The first three PSOs (GPSO, LPSO with ring neighborhood and VPSO with von

Neumann neighborhood) are regarded as standard PSOs and have been widely used in PSO

applications. The FIPS is a ‘fully informed’ PSO that uses all the neighbors to influence the

flying velocity. In FIPS, the URing topology structure is implemented with a wFIPS

algorithm for higher successful ratio. The HPSO-TVAC is a ‘performance-improvement’

PSO by improving the acceleration parameters and incorporating a self-organizing technique.

The DMS-PSO is devoted to improve the topological structure in a dynamic way. Finally, in

Table 2-2, the CLPSO offers a comprehensive learning strategy, aiming at yielding better

performance for multimodal functions. The parameter configurations for these PSO variants

are also given in Table 2-2, according to their corresponding references. In the tests, the

algorithm configuration of the APSO is as follows. The inertia weight ω is initialized to 0.9

and c1 and c2 to 2.0, same as the common configuration in a standard PSO. These parameters

are then adaptively controlled during the run. Parameter δ in (2-6) is a random value

uniformly generated in the interval [0.05, 0.1], while parameters σ in (2-9) linearly decreases

from σmax = 1.0 to σmin = 0.1.


41

For a fair comparison among all the PSO algorithms, they are tested using the same

population size of 20, a value of which is commonly adopted in PSO [163]. Further, all the

algorithms use the same number of 2.0×105 FEs for each test function [159]. All the

experiments are carried out on the same machine with a Celeron 2.26 GHz CPU, 256 MB

memory and the Windows XP2 operating system. For the purpose of reducing statistical

errors, each function is simulated 30 times independently and their mean results are used in

the comparison

2.4.2 Comparisons on the Solution Accuracy

The performance on the solution accuracy of every PSO listed in Table 2-3 is compared

with the APSO. The results are shown in Table 2-3, in terms of the mean and standard

deviation of the solutions obtained in the 30 independent runs by each algorithm. Boldface in

the table indicates the best result among those obtained by all eight contenders. Fig. 2-6

presents the comparison graphically in terms of convergence characteristics of the

evolutionary processes in solving the 12 different problems. Table 2-3 Results Comparisons on Solution Accuracy Among 8 PSOs on 12 Test Functions

Function GPSO LPSO VPSO FIPS HPSO-TVAC DMS-PSO CLPSO APSO

f1 Mean 1.98×10-53 4.77×10-29 5.11×10-38 3.21×10-30 3.38×10-41 3.85×10-54 1.89×10-19 1.45×10-150

Std. Dev 7.08×10-53 1.13×10-28 1.91×10-37 3.60×10-30 8.50×10-41 1.75×10-53 1.49×10-19 5.73×10-150

f2 Mean 2.51×10-34 2.03×10-20 6.29×10-27 1.32×10-17 6.9×10-23 2.61×10-29 1.01×10-13 5.15×10-84

Std. Dev 5.84×10-34 2.89×10-20 8.68×10-27 7.86×10-18 6.89×10-23 6.6×10-29 6.51×10-14 1.44×10-83

f3 Mean 6.45×10-2 18.60 1.44 0.77 2.89×10-7 47.5 395 1.0×10-10

Std. Dev 9.46×10-2 30.71 1.55 0.86 2.97×10-7 56.4 142 2.13×10-10

f4 Mean 28.1 21.8627 37.6469 22.5387 13 32.3 11 2.84

Std. Dev 24.6 11.1593 24.9378 0.310182 16.5 24.1 14.5 3.27

f5 Mean 0 0 0 0 0 0 0 0

Std. Dev 0 0 0 0 0 0 0 0

f6 Mean 7.77×10-3 1.49×10-2 1.08×10-2 2.55×10-3 5.54×10-2 1.1×10-2 3.92×10-3 4.66×10-3

Std. Dev 2.42×10-3 5.66×10-3 3.24×10-3 6.25×10-4 2.08×10-2 3.94×10-3 1.14×10-3 1.7×10-3

f7 Mean -10090.16 -9628.35 -9845.27 -10113.8 -10868.57 -9593.33 -12557.65 -12569.5

Std. Dev 495 456.54 588.87 889.58 289 441 36.2 5.22×10-11

f8 Mean 30.7 34.90 34.09 29.98 2.39 28.1 2.57×10-11 5.8×10-15

Std. Dev 8.68 7.25 8.07 10.92 3.71 6.42 6.64×10-11 1.01×10-14

f9 Mean 15.5 30.40 21.33 35.91 1.83 32.8 0.167 4.14×10-16

Std. Dev 7.4 9.23 9.46 9.49 2.65 6.49 0.379 1.45×10-15

f10 Mean 1.15×10-14 1.85×10-14 1.4×10-14 7.69×10-15 2.06×10-10 8.52×10-15 2.01×10-12 1.11×10-14

Std. Dev 2.27×10-15 4.80×10-15 3.48×10-15 9.33×10-16 9.45×10-10 1.79×10-15 9.22×10-13 3.55×10-15

f11 Mean 2.37×10-2 1.10×10-2 1.31×10-2 9.04×10-4 1.07×10-2 1.31×10-2 6.45×10-13 1.67×10-2

Std. Dev 2.57×10-2 1.60×10-2 1.35×10-2 2.78×10-3 1.14×10-2 1.73×10-2 2.07×10-12 2.41×10-2

f12 Mean 1.04×10-2 2.18×10-30 3.46×10-3 1.22×10-31 7.07×10-30 2.05×10-32 1.59×10-21 3.76×10-31

Std. Dev 3.16×10-2 5.14×10-30 1.89×10-2 4.85×10-32 4.05×10-30 8.12×10-33 1.93×10-21 1.2×10-30


42

An interesting result is that all PSO algorithms have most reliably found the minimum of

f5. It is a region rather than a point in f5 that is the optimum. Hence this problem may be

relatively easy to solve with a 100% success rate. The comparisons in both Table VI and Fig.

11 show that, when solving unimodal problems, the APSO offers the best performance on

most test functions. In particular, the APSO offers the highest accuracy on functions f1, f2, f3,

f4 and f5, and ranks third on f6.

Fig. 2-6 Convergence performance of the 8 different PSOs on the 12 test functions.

The APSO also achieves the global optimum on the optimization of complex

multimodal functions f7, f8, f9, f10 and f12. Although the CLPSO outperforms the APSO and

others on f11 (Griewank’s function), its mean solutions on other functions are worse than

those of the APSO. Further, the APSO can successfully jump out of local optima on most of

the multimodal functions and surpasses all other algorithms on functions f7, f8 and f9 where

the global optimum of f7 (Schwefel’s function) is far away from any of the local optima, and


43

the globally best solutions of f8 and f9 (continuous/noncontiguous Rastrigin’s functions) are

surrounded by a large number of local optima. The ability of avoiding being trapped into

local optima and achieving global optimal solutions to multimodal functions suggests that the

APSO can indeed benefit from the ELS.

2.4.3 Comparisons on the Convergence Speed

The speed in obtaining the global optimum is also a salient yardstick for measuring

algorithm performance. Since PSO is a population-based iterative algorithm, it is a common

to use the required function evaluation (FEs) times to achive an accepted solution under a

given threshold to measure the convergence rate of the algorithm. Table 2-4 reports the mean

FEs needed and mean CPU time needed to obtain an accepted solution given the threshold in

Table 2-4 for the algorithms on the 12 functions in 30 runs. The successful rate of each

algorithm is also demonstrated in Table 2-4. Note that, the mean FEs and mean CPU time are

calculated on the successful runs. For example, when the algorithm solves a function, if the

algorithm achives accepted solutions in only 27 out of 30 runs, then the successful rate of the

algorithm is 90%, and the mean FEs and mean CPU time is the average value of the 27 runs. Table 2-4 Convergence Speed and Algorithm Reliability Comparisons

Function GPSO LPSO VPSO FIPS HPSO-TVAC DMS-PSO CLPSO APSO

f1 Mean FEs 105695 118197 112408 32561 30011 91496 72081 7074 Time (sec) 0.96 1.12 1.04 0.36 0.29 0.85 0.48 0.11 Ratio (%) 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0


f3 Mean FEs 137985 162196 147133 73790 102499 185588 - 21166 Time (sec) 2.16 2.69 2.33 1.35 1.67 2.91 - 0.98 Ratio (%) 100.0 96.7 100.0 100.0 100.0 86.7 0.0 100.0



f6 Mean FEs 165599 161784 170675 47637 - 180352 99795 78117 Time (sec) 1.57 1.65 1.73 0.56 - 1.74 0.72 1.30 Ratio (%) 80.0 26.7 43.3 100.0 0.0 40.0 100.0 100.0




44





Avg. Reliability 88.62% 83.34% 85.56% 95.83% 88.06% 83.62% 91.67% 97.23%

Due to the adaptive control of the parameter and strategy based on the evolutionary state

of the algorithm, the algorithm can optimize the problem using different suitable parameters

during different evolutionary stages and converge to global optimum quickly. Table 2-4

reveals that the APSO generally offers a much higher speed, measured by either the mean

number of FEs or by the mean CPU time needed to reach an acceptable solution. The CPU

time is important to measure computational load, as many existing PSO variants have added

extra operations that cost computational time. Although the APSO needs to calculate the

mean distance between every pair of particles in the swarm, the calculation costs negligible

CPU time.

In solving real-world problems, the ‘function evaluation’ time overwhelms algorithm

overhead. Hence the mean number of FEs needed to reach the acceptable accuracy would be

much more interesting than the CPU time. Thus the mean FEs are also explicitly presented

and compared in Table 2-4. For example, tests on f1 show that the average numbers of FEs of

105695, 118197, 112408, 32561, 30011, 91496 and 72081 are needed by the GPSO, LPSO,

VPSO, FIPS, HPSO-TVAC, DMS-PSO and CLPSO algorithms, respectively, in order to

reach an acceptable solution. However, the APSO uses only 7074 FEs, while its CPU time of

0.11 second is also the shortest among the eight algorithms. In summary, the APSO uses the

least CPU time and the smallest number of FEs to reach acceptable solutions on 9 out of 12

test functions (f1, f2, f3, f4, f5, f7, f8, f9 and f11)

2.4.4 Comparisons on the Algorithm Reliability

Table 2-4 also reveals that the APSO offers a generally highest percentage of trials

reaching acceptable solutions and the highest reliability averaged over all the test functions.

The APSO reaches the acceptable solutions with a successful ratio of 100% on all the test


45

functions except function f11. Note that the HPSO-TVAC and the CLPSO did not converge

on functions f6 and f3, respectively. For the mean reliability of all the test functions, APSO

offers the highest reliability of 97.25%, followed by FIPS, CLPSO, GPSO, HPSO-TVAC,

VPSO, DMS-PSO and LPSO.

According to the theorem of “no free lunch” [164], one algorithm cannot offer better

performance than all others on every aspect or on every kind of problems. This is also

observed in our experimental results. The GPSO outperforms local version PSOs, including

the LPSO, VPSO and FIPS with the U-Ring structure, on simple unimodal functions f1, f2,

and f3. However, on difficult unimodal functions (e.g., the Rosenbrock’s function, f4) and the

multimodal functions, the LPSO and FIPS offer better performance than GPSO. The FIPS

achieves the highest accuracy on function f10 while CLPSO and DMS-PSO perform best on

f11 and f12, respectively, but these global algorithms sacrifice performance on unimodal

functions. However, the APSO outperforms most on both unimodal and multimodal functions,

owing to its adaptive parameters that deliver faster convergence and to its adaptive elitist

learning strategy that avoids local optima. Further, such outperformance has been achieved

with the highest success rate on all but Griewank’s function (f11).

Fig. 2-7 Cumulative percentages of the acceptable solutions obtained duiring the evolutionary process.


46

In order to depict how fast the algorithms reach acceptable solutions, accumulative

percentages of the acceptable solutions obtained in each function evaluation are shown in Fig.

2-7. The figure includes the representative unimodal functions (f1, f4) and complex

multimodal functions (f7, f8). For example, Fig. 2-7(c) shows that while optimizing function f7,

(i) the APSO, the CLPSO and the HPSO-TVAC manage to obtain acceptable solutions in all

the trials, but the APSO is faster than the CLPSO and the HPSO-TVAC; (ii) about only

two-thirds of the trails in the GPSO and the FIPS obtain acceptable solutions (with a medium

convergence speed); (iii) the VPSO succeeds in about 40% of the trials; (iv) the DMS-PSO

and the LPSO converge slowest and only succeed in about one-sixth of the trails.

2.4.5 Comparisons Using t-Tests

For a thorough comparison, the t-test [162] has also been carried out. Table 2-5 presents

the t values and P values on every function of this two-tailed test with a significance level of

0.05 between the APSO and another PSO algorithm. Rows “1 (Better)”, “0 (Same)” and “–1

(Worse)” give the number of functions that the APSO performs significantly better than,

almost the same as, and significantly worse than the compared algorithm, respectively. Row

“General Merit” shows the difference between the number of 1’s and the number of –1’s,

which is used to give an overall comparison between the two algorithms. For example,

comparing the APSO and the GPSO, the former outperformed the latter significantly on 7

functions (f2, f3, f4, f6, f7, f8 and f9), does as better as the latter on 5 functions (f1, f5, f10, f11 and

f12) and does worse on 0 function, yielding a “General Merit” figure of merit of 7 – 0 = 7,

indicating that the APSO generally outperforms the GPSO. Although it performed slightly

weaker on some functions, the APSO in general offered much improved performance than all

the PSOs compared, as confirmed by Table 2-5. Table 2-5 Comparisons Between the APSO and Other PSOs on t-Tests

PSOs Function GPSO LPSO VPSO FIPS HPSO-TVAC DMS-PSO CLPSO

f1 t-value 1.52851 2.31098† 1.4671 4.88501† 2.17917† 1.20579 6.93676†

P-value 0.13182 0.02441 0.14775 0.00001 0.03339 0.23279 0.00000

f2 t-value 2.35366† 3.85389† 3.96641† 9.17296† 5.48133† 2.16682† 8.47224†

P-value 0.02200 0.00029 0.00020 0.00000 0.00000 0.03437 0.00000

f3 t-value 3.73355† 3.3183† 5.08843† 4.8526† 5.32372† 4.60526† 15.24494†

P-value 0.00043 0.00157 0.00000 0.00001 0.00000 0.00002 0.00000

f4 t-value 5.57538† 8.96094† 7.58036† 32.85535† 3.29589† 6.62263† 3.00317†

P-value 0.00000 0.00000 0.00000 0.00000 0.00168 0.00000 0.00394

f5 t-value 0 0 0 0 0 0 0P-value 0 0 0 0 0 0 0

f6 t-value 5.76807† 9.46838† 9.26301† -6.35398† 13.34007† 8.08689† -1.98019


47

P-value 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.05243

f7 t-value 27.42668† 35.28576† 25.33892† 15.12005† 32.28794† 36.92784† 1.79505 P-value 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.07786

f8 t-value 19.3625† 26.38238† 23.13992† 15.03442† 3.52542† 24.02031† 2.12311 P-value 0.00000 0.00000 0.00000 0.00000 0.00083 0.00000 0.03802

f9 t-value 11.47452† 18.03982† 12.35315† 20.72115† 3.7845† 27.68977† 2.40832†

P-value 0.00000 0.00000 0.00000 0.00000 0.00037 0.00000 0.01923

f10 t-value 0.46159 6.73442† 3.78212† -5.12379† 1.19682 -3.58847† 11.89982†

P-value 0.64610 0.00000 0.00037 0.00000 0.23624 0.00068 0.00000

f11 t-value 1.08486 -1.07192 -0.70658 -3.56237† -1.23569 -0.6606 -3.79146†

P-value 0.28247 0.28820 0.48265 0.00074 0.22155 0.51148 0.00036

f12 t-value 1.79505 1.87465 1 -1.15597 8.68004† -1.61897 4.51261†

P-value 0.07786 0.06588 0.32146 0.25243 0.00000 0.11088 0.00003 1 (Better) 7 9 8 7 9 7 8 0 (Same) 5 3 4 2 3 4 3

-1 (Worse) 0 0 0 3 0 1 1 General Merit 7 9 8 4 9 6 7

2.5 Further Analysis of APSO

2.5.1 Analysis of Parameter Adaptation and Elitist Learning

APSO operations involve an acceleration rate δ in equation (2-6) and an elitist learning

rate σ in (2-9). Hence, are these new parameters sensitive in the operations? What impacts do

the two operations of parameter adaptation and elitist learning have on the performance of the

APSO? This section aims to answer these questions by further testing the APSO on 3

unimodal (f1, f2 and f4) and 3 multimodal (f7, f8 and f10) functions

In order to quantify the significance of these two operations, the performance of the

APSO without parameter adaptation or elitist learning is tested under the same running

conditions as in Section 2.4.1. Results of the mean values on 30 independent trials are

presented in Table 2-6. Table 2-6 Merits of Parameter Adaptation and Elitist Learning on Search Quality

Algorithm APSO With Both Adaptation & Learning

APSO With Only Adaptive Parameters

APSO With Only ELS

GPSO (PSO Without Either)

Function Mean Std. Dev Mean Std. Dev Mean Std. Dev Mean Std. Dev f1 1.45×10-150 5.73×10-150 7.67×10-160 3.42×10-159 3.6×10-50 1.43×10-49 1.98×10-53 7.08×10-53

f2 5.15×10-84 1.44×10-83 6.58×10-88 2.34×10-87 2.41×10-32 9.98×10-32 2.51×10-34 5.84×10-34

f4 2.84 3.27 13.8879 14.6335 12.7464 18.1979 28.0972 24.5981 f7 -12569.5 5.22×10-11 -7367.77 681.983 -12569.5 3.34×10-12 -10090.16 495.135 f8 5.8×10-15 1.01×10-14 52.7327 15.0326 1.78×10-16 5.42×10-16 30.6779 8.6781 f10 1.11×10-14 3.55×10-15 1.0885 1.00363 1.12×10-14 2.64×10-15 1.15×10-14 2.27×10-15


48

It is clear from the results that with elitist learning alone and without adaptive control of

parameters, the APSO can still deliver good solutions to multimodal functions. However, the

APSO suffers from lower accuracy in solutions to unimodal functions. As algorithms can

easily locate the global optimal region of a unimodal function and then refine the solution, the

lower accuracy may be caused by the slower convergence speed to reach the global optimal

region. On the other hand, the APSO with parameters adaptation alone but without ELS can

hardly jump out of local optima and hence results in poor performance on multimodal

functions. However, it can still solve unimodal problems well.

Note that both reduced APSO algorithms generally outperform a standard PSO that

involves neither adaptation parameters nor elitist learning. However, the full APSO is the

most powerful and robust for any tested problem. This is most evident in the test results on f4.

These results confirm the hypothesis that parameters adaptation speeds up the convergence of

the algorithm and elitist learning helps the swarm jump out of local optima and find better

solutions.

2.5.2 Search Behaviors of APSO and Parameter Evolution Analysis

In order to understand the running mechanisms of APSO further, we first investigate its

search behaviors and parameter variation on unimodal function Sphere function f1 and

multimodal function Rastrigin function f8.

Firstly, the test is performed on the unimodal function f1. In a unimodal space, it is

important for an optimization or search algorithm to converge fast and to refine the solution

for a high accuracy. The inertia weight shown in Fig. 2-8(a) confirms that the APSO

maintains a large ω in the exploration phase (for about 50 generations) and then a rapidly

decreasing ω follows exploitation leading to convergence, as the unique global optimum

region is found by a leading particle and the swarm follows it.

Fig. 2-8(b) shows how ESE in the APSO has influenced the acceleration coefficients.

The curves for c1 and c2 somewhat show good agreement with the ones given in Fig. 2-5. It

can be seen that c1 increases whilst c2 decreases in the exploration and exploitation phases.

Then c1 and c2 reverse their directions when the swarm converges, eventually returning to

around 2.0. Then trials in elitist learning perturb the particle that leads the swarm, which is

reflected in the slight divergence between c1 and c2 that follows. The search behavior on the

unimodal function indicates that the proposed APSO algorithm has indeed identified the

evolutionary states and can adaptively control the parameters for improved performance.


49

Fig. 2-8 Search behaviors of the APSO on Sphere function: (a) Mean value of ω during the run time

showing an adaptive momentum; (b) Mean values of c1 and c2 adapting to the evolutionary states.

Secondly, test is carried out on f8. Here the APSO is tested again to see how it will adapt

itself to a multimodal space. When solving multimodal functions, a search algorithm should

maintain diversity of the population and search for as many optimal regions as possible. The

search behavior of the APSO is investigated on the Rastrigin function (f8 in Table 2-1). In

order to compare the diversity in the search by the APSO and the traditional PSO, a yardstick

proposed in [165] is used here called the ‘population standard deviation’, denoted by psd

)1/(])([1 1

2 −−= ∑ ∑= =Nxxpsd N

i

D

j

jji (2-10)

where N, D and x are the population size, the number of dimension and the mean position

of all the particles, respectively.

The variations in psd can indicate the diversity level of the swarm. If psd is small, it

indicates that the population has converged closely to a certain region, and the diversity of the

population is low. A larger value of psd indicates that the population is of a higher diversity.

However, it does not necessarily mean that a larger psd is always better than a smaller one

because an algorithm which cannot converge may also present a large psd. Hence, the psd

needs to be considered together with the solution that the algorithm arrives at.

Results of psd comparisons are plotted in Fig. 2-9(a) and those of the evolutionary

processes in Fig. 2-9(b). It can be seen that the APSO has an ability to jump out of local

optima, reflected by the regained diversity of the population, as revealed in Fig. 2-9(a), with a

steady improvement in the solution, as shown in Fig. 2-9(b). Fig. 2-9(c) and (d) show the

inertia weight and the acceleration coefficients behaviors of the APSO, respectively. These

plots confirm that, in a multimodal space, the APSO can also find a potential optimal region

(maybe a local optimum) fast in an early phase and converge fast with a rapid decreasing

diversity, due to the adaptive parameters strategies. However, if a current optimal region is


50

local, the swarm can separate and jump out. Hence, the APSO can appropriately increase the

diversity of the population so as to explore for a better region, owing to the ELS in the

convergence state. This behavior with adaptive population diversity is valuable for a global

search algorithm to prevent from being trapped in local optima and to find the global

optimum in a multimodal space

Fig. 2-9 Search behaviors of PSOs on Rastrigin’s function: (a) Mean psd during the run time; (b) Plots of

convergence during the minimization run; (c) Mean value of the ω during the run time showing an

adaptive momentum; (d) Mean values of c1 and c2 adapting to the evolutionary states.

2.5.3 Sensitivity of the Acceleration Rate

APSO introduce two new parameters, δ in Eq. (2-6) andσ in Eq. (2-9). This section

and the next section will analyze the sensitivity of these two parameters, respectively.

The effect of the acceleration rate, reflected by its bound δ, on the performance of the

APSO is investigated here. For this, the learning rate σ is hence fixed (e.g., σmax=σmin=0.5)

and the other parameters of the APSO remain the same as in Chapter 2.4.1. The investigation

consists of 6 test strategies for δ, the first 3 being to fix its value to 0.01, 0.05 and 0.1,

respectively, and the remaining 3 being randomly to generate its value using a uniform

distribution within [0.01, 0.05], [0.05, 0.1] and [0.01, 0.1], respectively. The results are

presented in Table 2-7, in terms of the mean of the solutions found in 30 independent trials.


51

Table 2-7 Effects of the Acceleration Rate on Global Search Quality

Value of f1 f2 f4 f7 f8 f10 Fixed at 0.01 6.79×10-151 1.23×10-83 3.4159 -12474.3 2.25×10-2 1.08×10-14

Fixed at 0.05 8.3×10-149 2.81×10-83 4.06522 -12317 6.16×10-15 1.14×10-14

Fixed at 0.1 8.03×10-149 6.05×10-84 3.72976 -12153.8 6.63×10-2 1.14×10-14

Random (0.01,0.05) 2×10-148 2.15×10-80 2.6106 -12420.7 0.132661 1.14×10-14

Random (0.05,0.1) 2.62×10-150 6.95×10-82 1.79069 -12475.2 8.11×10-15 1.17×10-14

Random (0.01,0.1) 2.64×10-149 3.12×10-83 3.00886 -12133.6 9.95×10-2 1.11×10-14

It can be seen that the APSO is not very sensitive to the acceleration rate δ and the six

acceleration rates all offer good performance. This may be owing to the use of bounds for the

acceleration coefficients and the saturation to restrict their sum by (2-7). Therefore, given the

bounded values of c1 and c2 and their sum restricted by (2-7), an arbitrary value within the

range [0.05, 0.1] for δ should be acceptable to the APSO algorithm

2.5.4 Sensitivity of the Elitist Learning Rate

In order to assess the sensitivity of σ in elitist learning, six strategies for setting its value

are tested here, using 3 fixed values (0.1, 0.5 and 1.0) and 3 time-varying ones (from 1.0 to

0.5, from 0.5 to 0.1, and from 1.0 to 0.1). All other parameters of the APSO remain as those

in Section 2.4.1. The mean results of 30 independent trials are presented in Table 2-8. Table 2-8 Effects of the Elitist Learning Rate on Global Search Quality

Value of f1 f2 f4 f7 f8 f10 Fixed at 0.1 5.16×10-152 1.62×10-82 1.94812 -11622 6.87×10-15 1.10×10-14

Fixed at 0.5 6.83×10-148 7.02×10-77 1.73717 -12045.9 3.32×10-2 1.05×10-7 Fixed at 1.0 2.07×10-149 3.39×10-83 2.7744 -12277.3 9.95×10-2 1.12×10-14

From 1.0 to 0.5 2.85×10-148 5.21×10-82 2.34211 -12263.9 0.132661 1.34×10-14

From 0.5 to 0.1 1.90×10-148 8.83×10-82 2.0236 -12565.5 6.63×10-2 1.21×10-14

From 1.0 to 0.1 1.24×10-152 8.71×10-83 3.23075 -12569.5 4.03×10-15 1.12×10-14

The results show that if σ is small (e.g., 0.1), the learning rate is not enough for a long

jump out of local optima, evident in the performance on f7. However, all other settings, which

permit a larger σ, have delivered almost the same excellent performance, especially the

strategy with a time-varying σ decreasing from 1.0 to 0.1. It is seen that a smaller σ

contributes more to helping the leading particle refine, while a larger σ contributes more to

helping the leader move away from its existing position so as to jump out of local optima.

This confirms the intuition that long jumps should be accommodated at an early phase to

avoid local optima and premature convergence, whilst small perturbations at a latter phase

should help refine global solutions, as recommended in this chapter.


52

2.6 Chapter Summary

This chapter introduced the statistical analysis technique into particle optimization

algorithm, and proposed an adaptive particle optimization algorithm (APSO). Due to the

online perception and analysis process ability on the population distribution and fitness data

of the statistical analysis technique, APSO defines an evolutionary factor to describe the

evolutionary state. With the identification and classification ability of the evolutionary factor,

APSO adaptively control the parameters and strategies. As shown in the benchmark tests, the

adaptive control of the inertia weight and the acceleration coefficients makes the algorithm

extremely efficient, offering a substantially improved convergence speed, in terms of both

number of FEs and CPU time needed to reach acceptable solutions for both unimodal and

multimodal functions.

The features and advantages of APSO are:

1) Combine the statistical analysis technique with PSO algorithm. Through the

perception and analysis of the population distribution information and relative particle fitness,

this chapter proposed an adaptive PSO. Machine learning aided method is an effective way to

design adaptive PSO or other adaptive EAs.

2) Adaptive control makes the algorithm evolve suitanl strategies and parameter values as

evolution progresses in differwnt evolutionary states and performs rapid convergence

capacity and global searching ability.

3) APSO remains the simplicity, easy implement, efficiency of the original PSO. The

introduction of the evolutionary state estimation and parameter and strategy adaptive control

process, which is based on the simple and efficient statistical analysis technique, requires

little computation consumption.Tthe APSO is still simple and almost as easy to use as the

standard PSO, whilst it brings in substantially improved performance in terms of convergence

speed, global optimality, solution accuracy, and algorithm reliability. Moreover, the

acceleration rate and learning rate have insignificant impact on the performance of the PSO,

which enhance the accessibility of the APSO.

In a conclusion, with the help of statistical analysis technique, this chapter proposed an

adaptive particle swarm optimization to accelerate convergence speed and enhance solution

accuracy. It is an important and successful exploration of machine learning aided particle

swarm optimization algorithm.

Chapter 3 Orthogonal Learning Particle Swarm Optimization Based on Orthogonal Experiments Design Techique in Machine Learning

53

Chapter 3 Orthogonal Learning Particle Swarm Optimization

Based on Orthogonal Experiments Design Techique in Machine

Learning

3.1 Introduction

The salient feature of PSO lies in its learning mechanism that distinguishes the

algorithm from other EC techniques. When searching for a global optimum in a hyperspace,

particles in a PSO fly in the search space according to guiding rules. It is the guiding rules

that make the search effective and efficient. In the traditional PSO, the rules are the

mechanism that each particle learns from its own best historical experience pBesti and its

neighborhood’s best historical experience nBesti. But how to make full use of these guidance

information to bring better learning efficiency to PSO and hence better global optimization

performance is an impotant and challenging problem.

As described in Chapter 1.2.4, according to the method of choosing the neighborhood’s

best historical experience, PSO algorithms are traditionally classified into global version PSO

(GPSO) and local version PSO (LPSO). Without loss of generality, this chapter aims at

improving the performance of both the GPSO and the LPSO with the ring structure, where a

particle takes its left and right particles (by particle index) as its neighbors. In both GPSO and

LPSO, the information of a particle’s best experience and its neighborhood’s best experience

is utilized in a simple way, where the flying is adjusted by a simple learning summation of

the two experiences which will be given in Eq. (1-6). However, this is not necessarily an

efficient way to make the best use of the search information in these two experiences. For

example, in one case, it may cause an ‘oscillation’ phenomenon because the guidance of the

two experiences may be in opposite directions. This is inefficient to the search ability of the

algorithm and delays the convergence speed. In another case, the particle may suffer from the

‘two steps forward, one step back’ phenomenon that some components of the solution vector

may be improved by one exemplar but may be deteriorated by the other. This is because that

one exemplar may have good values on some dimensions of the solution vector while the

other exemplar may have good values on some other dimensions. Hence, how to discover

more useful information embedded in the two exemplars and thus how to utilize the

information to construct an efficient and promising exemplar to guide the particle flying


54

steadily towards the global optimal region are important and challenging research issues that

PSO researchers need to pay attention to.

The ‘oscillation’ phenomenon is likely to be caused by linear summation of the personal

influence and the neighborhood influence. For the clearness and easiness of understanding,

we first simplify the Eq. (1-6) as Eq. (3-1) by removing the inertia weight component and the

random values.

Vid = (Pid – Xid) + (Nid – Xid) (3-1)

Fig. 3-1 The “oscillation” pheronomon caused by traditional PSO learning strategy.

In Eq. (3-1), we consider the following case for a maximization problem where the

current particle Xi is between its personal best position pBesti and its neighborhood’s best

position nBesti, as shown in Fig. 3-1. At first, the distance between nBesti and Xi may be

farther than the one between pBesti and Xi, as in Fig. 3-1 (a), then Xi will move towards

nBesti because of its larger pull. However, as moving towards nBesti, the distance between

pBesti and Xi will increase, as shown in Fig. 3-1 (b). In this case, the particle will move

towards pBesti instead. The oscillation would thus occur and the particle will be puzzled in

deciding where to stay. This oscillation phenomenon causes inefficiency to the search ability

of the algorithm and delays convergence.

Another related phenomenon of the traditional learning mechanism is the ‘two step

forward, one step back’ phenomenon as described in Fig. 3-2. For example, given a

3-dimension Sphere function 23

22

21)( xxxf ++=X , whose global minimum point is [0, 0, 0].

Suppose that the current position is Xi = [2, 5, 2], its personal best position is pBesti = [0, 2, 5]

and its neighborhood’s best position is nBesti = [5, 0, 1]. The updated velocity is Vi = [1, –8,

2] according to Eq. (3-1), and thus the new position is Xi = Xi + Vi = [3, –3, 4], resulting in a

new position with a cost value of 34 which is worse than Xi and pBesti. Therefore, the

particle does not benefit from the learning from pBesti and nBesti in this generation. However,


55

vectors pBesti and nBesti indeed possess good information in their structures. For example, if

we can discover good dimensions of the two vectors, we can then combine them to form a

new guidance vector of oBesti = [0, 0, 1] where the first coordinate 0 comes from pBesti

while the second and the third coordinates 0 and 1 come from nBesti (with corresponding

dimension). Given the guidance of Po, the updated velocity become Vi = oBesti – Xi = [0, 0, 1]

– [2, 5, 2] = [–2, –5, –1]; thus the new position is Xi = Xi + Vi = [0, 0, 1], resulting in a new

and better position with a cost f(Xi) = 1 that makes the particle fly faster towards the global

optimum [0, 0, 0].

[2,5, 2]( ) ( ) [ 2, 3,3] [3, 5, 1] [1, 8, 2]

[0, 2,5][2,5, 2] [1, 8,2] [3, 3, 4]

[5,0,1]

= ⎫= − + − = − − + − − = −⎪= ⇒⎬ = + = + − = −⎪= ⎭

ii i i i i

ii i i

i

XV pBest X nBest X

pBestX X V

nBest

(a) Traditional PSO learning strategy

[2,5,2] [0,2,5]( ) [ 2, 5, 1]

[0,2,5] [0,0,1][0,0,1]

[5,0,1] [5,0,1]

= ⎫= − = − − −⎪= ⇒ = ⊕ = ⊕ = ⇒⎬ = + =⎪= ⎭

ii i i

i i i ii i i

i

XV oBest X

pBest oBest pBest nBestX X V

nBest(b) Orthogonal PSO learning strategy

Fig. 3-2 The “two steps forward, one step back” pheronomon caused by traditional PSO learning strategy.

It follows that traditional PSO is not able to perform efficient global searching not just

because the particle can not find high-quality position, but because the traditional learning

strategy can not make full use of the information in the found solutions. Although the

solutions obtained in the evolutionary process are not enough good, these solutions have good

structures generally. Especially in the process of learning from its own best historical

experience pBesti and its neighborhood’s best historical experience nBesti, how to find more

useful information from these two learning exemplars and then construct a better learning

exemplars to guide the fly of the particle is important. Machine learning technique is artificial

intelligence technology and is able to discover useful information from the available data to

guide the actions of the particle. In order to make full use of the search information of both

pBesti and nBesti, this chapter introduced the orthogonal prediction technique in machine

learning into PSO and construct a learning vector to guide the particle fly towards global

optimal position more steadily.

Because orthogonal experimental design (OED) offers an ability to discover the best

combination levels for different factors with a reasonably small number of experimental

samples, in this chapter, we propose to use the OED method to construct a promising learning

exemplar. Here, OED is used to discover the best combination oBesti (Orthogonal Best) of a


56

particle’s best historical position pBesti and its neighborhood’s best historical position nBesti.

The orthogonal experimental factors are the dimensions of the problem and the levels of each

dimension (factor) are the two choices of a particle’s best position value and its

neighborhood’s best position value on this corresponding dimension. If we exhaustively test

all the combinations of pBesti and nBesti for the best guidance vector oBesti, 2D trials are

need. OED has high orthogonal test ability and prediction ability. Through testing a few

representative combinations with typical orthogonal combinations method and predicting the

potentially best combinations with a factor analysis (FA) method, ODE is able to discover

useful information from pBesti and nBesti.This way, the best combination of the two

exemplars can be constructed to guide the particle to fly more steadily, rather than oscillatory,

because only one constructed exemplar is used for the guidance. It is thus expected for a

particle to fly more promisingly towards the global optimum because the constructed

exemplar makes the best use of the search information of both the particle’s best position and

its neighborhood’s best position.

Therefore, with the aid of orthogonal prediction technique in machine learning, this

chapter designs an orthogonal learning (OL) strategy, and proposes orthogonal learning

particle swarm optimization (OLPSO) based on the OL stategy. The important contributions

and innovations include 3 aspects as follow:

1) Introduce the orthogonal prediction technique in machine learning into PSO to

discover, analyze, and prodict the information of individuals and population, and to form an

orthogonal learning (OL) strategy to bring better learning efficiency to PSO and hence better

global optimization performance;

2) Different from the idea of using OED to construct a better solution for EA before,

such as using ODE to design crossover operator [166], initialize the searching space [167],

search local optima [168][169] in GA; and similar methods of using OED to find a better

solution are also used in SA [170][171], ACO [172], and PSO[150][151][149], the OL

strategy in this chapter is to discover, analyze, and predict the exiting searching information

to construct a vector to guide the algorithm searching. Therefore, the OL strategy in this

chapter is focused on designing a guidance exemplar with an ability to predict promising

search directions towards the global optimum but not finding out a better solution;

3) The OL strategy designed in this chapter is a generic operator and can be applied to

PSO with any kind of topology structure. It is applied on GPSO and LPSO to verify the

effectiveness and efficiency of OL strategy. The experimental results not only demonstrate


57

the advantage of OL strategy and the OLPSO algorithm, but also give a good inspiration of

how to apply the OL strategy to other PSO versions.

3.2 Orthogonal Learning Particle Swarm Optimization

3.2.1 Orthogonal Experimental Design

In order to illustrate how to use the OED, a simple example is shown in Table 3-1,

which arises from chemical experiments. In this example, the aim is to find the best level

combination of the 3 factors involved to increase the conversion ratio. Table 3-1 shows that 3

factors, which will affect experimental results, are the temperature, time and alkali, denoted

as factors A, B, and C, respectively. Moreover, there are 3 levels (different choices) involved

in each factor. For example, the temperature can be 80ºC, 85ºC, or 90ºC. Thus, there are in

total 33=27 combinations of experimental designs. This is a combinatorial explosion problem.

When the dimension increases, the possible combinations increases rapidly. So enumeration

is not suitable to address this problem. However, with the help of OED, one can obtain or

predict the best combination by testing only few representative experimental cases. Take the

example shown in Table 3-1 with the describtion in Table 3-2, we introduce the procedure of

OED method as follow: Table 3-1 The Factors and Levels of the Chemical Experiment Example

Factors Levels

A Temperature (0C)

B Time (Min)

C Alkali (%)

Level1 L1 80 90 5 Level2 L2 85 120 6 Level3 L3 90 150 7

49

1 1 1 11 2 2 21 3 3 32 1 2 3

(3 ) 2 2 3 12 3 1 23 1 3 23 2 1 33 3 2 1

L

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3-2)


58

Step 1, Combination test based on orthogonal array: The OED method works on a

predefined table called an orthogonal array (OA). An OA with N factors and Q levels per

factor is always denoted by LM(QN), where L denotes the orthogonal array and M is the

number of combinations of test cases. For the example shown in Table 3-1, the L9(34) OA

given by Eq.(3-2) is suitable.

The OA in (3-2) has 4 columns, meaning that it is suitable for the problems with at most

4 factors. As any sub columns of an OA is also an OA, we can to use only the first 3 columns

(or arbitrary 3 columns) of the array for the experiment. For example, the first three columns

in the first row is [1, 1, 1], meaning that in this experiment, the first factor (temperature), the

second factor (time), and the third factor (alkali) are all designed to the 1st level, that is, 80ºC,

90 minutes, and 5% as given in Table 3-1. Similarly, combination of [1, 2, 2] is used in the

second experiment, and so on. The total of 9 experiments specified by the L9(34) are

presented in Table 3-2. Table 3-2 Deciding the Best Combination Levels of the Chemical Experimental Factors Using an

Orthogonal Experimental Design Method

Combination A:Temperature(0C) B: Time (Min) C: Alkali (%) Result C1 (1) 80 (1) 90 (1) 5 F1=31 C2 (1) 80 (2) 120 (2) 6 F2=54 C3 (1) 80 (3) 150 (3) 7 F3=38 C4 (2) 85 (1) 90 (2) 6 F4=53 C5 (2) 85 (2) 120 (3) 7 F5=49 C6 (2) 85 (3) 150 (1) 5 F6=42 C7 (3) 90 (1) 90 (3) 7 F7=57 C8 (3) 90 (2) 120 (1) 5 F8=62 C9 (3) 90 (3) 150 (2) 6 F9=64

Level Factor Analysis L1 (F1+ F2+ F3)/3=41 (F1+ F4+ F7)/3=47 (F1+ F6+ F8)/3=45 L2 (F4+ F5+ F6)/3=48 (F2+ F5+ F8)/3=55 (F2+ F4+ F9)/3=57 L3 (F7+ F8+ F9)/3=61 (F3+ F6+ F9)/3=48 (F3+ F5+ F7)/3=48

Result of OED A3 B2 C2

Step 2, Prediction based on Factor Analysis: The ability of discovering the best

combination of levels is through the factor analysis (FA). The FA is based on the

experimental results of all the M cases of the OA. The FA results are shown in Table 3-2

and the process is described as follows. Let Fm denote the experimental result of the mth

( Mm ≤≤1 ) combination and Snq denote the effect of the qth ( Qq ≤≤1 ) level in the nth

( Nn ≤≤1 ) factor. The calculation of Snq is to add up all the Fm in which the level is q in

the nth factor, and then divide the total count of zmnq, as shown in Eq. (3-3) where zmnq is 1


59

if the mth experimental test is with the qth level of the nth factor, otherwise, zmnq is 0. For

example, to

1

1

Mm mnqm

nq Mmnqm

F zS

z=

=

×= ∑

∑ (3-3)

In this way, the effect of each level on each factor can be calculated and compared, as

shown in Table 3-2. For example, when we calculate the effect of level 1 on factor A,

denoted by element A1, the experimental results of C1, C2, and C3 are summed up for Eq.

(3-3) because only these combinations are involved in level 1 of factor A. Then the sum

divides the combination number (3 in this case) to yield Snq (SA1 in this case).

Step 3, Best combination: With all the Snq calculated, the best combination of the levels

can be determined by selecting the level of each factor that provides the highest-quality

Snq. For a maximization problem, the larger the Snq is, the better the qth level on factor n

will be. Otherwise, vice versa. As in the maximization example shown in Table 3-2, the

best result is the combination of A3, B2 and C2. Although the combination of (A3, B2,

C2) itself does not exist in the 9 combinations tested, it is discovered by the FA process.

3.2.2 Orthogonal Learning Strategy

Using the OED method, the original PSO can be modified as an orthogonal learning

PSO with an OL strategy that combines information of pBesti and nBesti to form a better

guidance vector oBesti = [Oi1, Oi2, …, OiD]. The particle’s flying velocity is thus changed as:

Vid = ωVid + crd(Oid – Xid) (3-4)

where ω is the same as in (1-4) that linearly decreases from 0.9 to 0.4, and c is fixed to be 2.0,

the same as c1 and c2, and rd is a random value uniformly generated within the interval [0, 1].

The guidance vector oBesti is constructed from pBesti and nBesti as (3-5):

oBesti = pBesti ⊕ nBesti (3-5)

where the symbol ⊕ stands for the OED operation. Therefore, the value oBestid comes from

pBesti or nBesti as the construct result of OED. With this efficient learning exemplar oBesti,

particle i adjusts its flying velocity, position and updates its personal best position in every

generation. In order to avoid the guidance changing the direction frequently, the vector oBesti

will be used as the exemplar for a certain number of generations until it cannot lead the

particle to a better position any more. For example, if the personal best position pBesti has

not been improved for G generations, then particle i will reconstruct a new oBesti by using


60

pBesti and nBesti. On the other hand, as oBesti is used for some time until it cannot improve

the position, one problem should be addressed is how to use the information that comes from

pBesti and nBesti immediately after pBesti and nBesti go to a better position during the search

process. In our implementations, vector oBesti stores only the index of pBesti and nBesti, not

the copy of the real position values. That is, oBestid only indicates that the dth dimension is

guided by pBesti or nBesti, it does not store the current value of pBestid or nBestid. Thus, in

the OLPSO algorithm, when pBesti or nBesti moves to a better position, the new information

will be used immediately by the particle through oBesti.

The construction process of oBesti is described as the following six steps:

Step 1: An OA is generated as LM(2D) where 2log ( 1)2 DM +⎡ ⎤⎢ ⎥= , using the procedure as

given follow:

1) Determine the row number ⎡ ⎤)1(log22 += DM , the column number N = M–1, and the

basic column number u=log2(M).

2) The elements in the basic columns are set as:

1[ ][ ] ( ) mod 22u k

aL a b −

−⎢ ⎥= ⎢ ⎥⎣ ⎦ (3-6)

where a=1, 2,…, M is the row index, b=2k–1 is the basic column index, and k=1, 2,…, u.

3) The elements in other columns are set as:

L[a][b+s] = (L[a][s] + L[a][b]) mod 2 (3-7)

where a=1, 2,…, M is the row index, b=2k–1 is the basic column index, s=1, 2, …, b–1,

and k=2,…, u.

4) For all the elements in the OA, transform the level value to 1 for the first level and

the level value to 2 for the second level.

Step 2: Make up M tested solutions Xj (1≤j≤M) by selecting the corresponding value

from pBesti or nBesti according to the OA. Here, if the level value in the OA is 1, then the

corresponding factor (dimension) selects pBesti; otherwise, selects nBesti.

Step 3: Evaluate each tested solution Xj (1≤j≤M), and record the best (with best fitness)

solution Xb.

Step 4: Calculate the effect of each level on each factor and determine the best level for

each factor using Eq. (3-3).

Step 5: Get a predictive solution Xp with the levels determined in Step 4 and evaluate Xp.

Step 6: Compare f(Xb) and f(Xp) and the level combination of the better solution is used

to construct the vector oBesti.


61

3.2.3 Orthogonal Learning Particle Swarm Optimization

The OL strategy is a generic operator and can be applied to any kind of topology

structure. If the OL is used for the global version PSO, then nBesti is gBest. If it is used for

the local version PSO, then nBesti is lBest. Either for a global or a local version, when

constructing the vector of oBesti, if pBesti is the same as nBesti (e.g., for the globally best

particle, pBesti and gBest are identical vectors), the OED makes no contribution. In such a

case, OLPSO will randomly select another particle pBestr, and then construct oBesti by using

the information of pBesti and pBestr through the OED.

The flowchart of OLPSO is shown in Fig. 3-3.

Start

Initialization Vi, Xi. Calculate pBesti, nBesti. Set gen=0, ω=0.9, c=2.0.

For each particle, construct the learning exemplar oBesti through pBesti and

nBesti.

gen<GENERATION

ω = 0.9–0.5 gen/GENERATION;i = 1;

For each dimension dvid = ω vid + c rd Bestid – xid)

xid = xid + vid

Xmin<=Xi<=Xmax

Evaluate particle i

f(Xi)<f(pBesti)

pBesti = Xistagnatedi = 0

f(pBesti)<f(nBesti)

nBesti = pBesti

i = i + 1;

i <= SIZE

stagnatedi = stagnatedi + 1;

stagnatedi > G

construct the learning exemplar oBesti through

pBesti and nBesti.

stagnatedi = 0;

gen = gen + 1;

Finish

No

Yes

Yes

Yes

Yes

YesYes

No

No

No

NoNo

Fig. 3-3 The flowchart of OLPSO.


62

3.3 Experimental Verification and Comparisons

3.3.1 Functions Tested

Sixteen benchmark functions listed in Table 3-3 are used in the experimental tests.

These benchmark functions are widely adopted in benchmarking global optimization

algorithms [59][162][163]. In this chapter, the functions are divided into three groups. The

first group includes 4 unimodal functions, where f1 and f2 are simple unimodal, f3

(Rosenbrock) is unimodal in a 2-dimensional or 3-dimensional search space but can be

treated as a multimodal function in high dimension cases [173], and f4 is with noisy

perturbation. The second group includes 6 complex multimodal functions with high

dimensionality. The last group includes 4 rotated multimodal functions and 2 shifted

functions defined in [163].

All functions are minimization problems with D=30 with the global optimal value of 0

except for f15 and f16 whose value are 390 and -330, respectively, due to the shift of the global

optimal value. Table 3-3 gives the global optimal solution (Column 5). Moreover, biased

initializations (Column 4) and the ‘Accept’ (Column 6) is also defined for each test function.

If a solution found by an algorithm falls between the acceptable value and the actual global

optimum fmin (Column 5), the run is judged to be successful. It should be note that f3, f8, f11,

f12, f13, f14 and f15 are coupling functions.

3.3.2 Compared Algorithm Configuration

Variant PSO algorithms, as detailed in Table 3-4, are used for comparisons. The

parameter configurations are all based on the suggestions in the corresponding references.

The first two are traditional PSOs of GPSO [45] and LPSO [54]. The third is a ‘fully

informed’ PSO (FIPS) [25] that uses all the neighbors to influence the flying velocity. The

fourth is a ‘performance-improvement’ PSO by improving the acceleration coefficients,

namely hierarchical PSO with time-varying acceleration coefficients (HPSO-TVAC) [28].

The fifth is a dynamic multi-swarm PSO (DMS-PSO) [57] which is designed to improve the

topological structure in a dynamic way. The sixth, CLPSO [59], aims to offer a better

performance for multimodal functions by using a CL strategy. The seventh, the OPSO [150]

algorithm, aims to improve the algorithm by using an OED to generate a better position, not

by constructing a learning exemplar as proposed in this chapter. These PSO variants are used


63

for comparisons because they are typical PSOs that are reported to perform well on their

studied problems. Moreover, they span a wide time interval from 1998 to 2008, which

witness the developments of PSO on variant aspects. For OLPSO developed in this chapter,

we implement the OL strategy in both the global and the local version PSO, resulting in two

OLPSO algorithms, the OLPSO-G and the OLPSO-L, respectively. Both will be compared

with GPSO, LPSO, FIPS, HPSO-TVAC, DMS-PSO, CLPSO, and OPSO. Table 3-3 Sixteen Test Functions Used in the Comparison

Type Test function Search Range

Initialization Range

Global Opt. x* Accept Name

Uni

mod

al

∑ == D

i ixxf1

21 )( [-100,100]D [-100,50]D {0}D 1×10-6 Sphere[162]

∑ ∏= =+= D

i

D

i ii xxxf1 12 )( [-10,10]D [-10,5]D {0}D 1×10-6 Schwefel’sP2[162]

1 2 2 23 11( ) [100( ) ( 1) ]D

i i iif x x x x−

+== − + −∑ [-10,10]D [-10,10]D {1}D 100 Rosenbrock[162]†

∑ =+= D

i i randomixxf1

44 )1,0[)( [-1.28,1.28]D [-1.28,0.64]

D {0}D 0.01 Noise[162]

Mul

timod

al

∑ =−×= Di ixixDxf 1 )sin(9829.418)(5 [-500,500]D [-500,500]D {420.96}D 2000 Schwefel[162]

∑ =+−= D

i ii xxxf1

26 ]10)2cos(10[)( π [-5.12,5.12]D [-5.12,2]D {0}D 100 Rastrigin[162]

exD

xD

xfD

i iD

i i ++−−−= ∑∑ ==20)2cos1exp( )12.0exp(20)(

112

7 π [-32,32]D [-32,16]D {0}D 1×10-6 Ackley [162]

∑ ∏= =+−= D

i

D

i ii ixxxf1 1

28 1)/cos(4000/1)( [-600,600]D [-600,200]D {0}D 1×10-6 Griewank [162]

∑

∑

=

+−

=

+−+

+−+=

D

i iD

iD

i i

xuy

yyyD

xf

12

1221

112

9

)4,100,10,(})1(

)](sin101[)1()(sin10{)( πππ

⎪⎩

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⎧

−<−−

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=++=

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axaaxaxk

mkaxuxy

im

i

i

im

i

iii

,)(

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),,,( ),1(41

1 where

[-50,50]D [-50,25]D {0}D 1×10-6

Generalized Penalized[162]

∑

∑

=

+−

=

++−+

+−+=

D

i iDD

iD

i i

xuxx

xxxxf

122

1221

112

10

)4,100,5,()]}2(sin1[)1(

)]3(sin1[)1()3({sin101)(

π

ππ [-50,50]D [-50,25]D {0}D 1×10-6

Rot

ated

and

Shi

fted

matrix orthogonalan is M ),96.420(*Mwherer

,96.420,otherwise ,0

500|| if ),||sin( where

1 ,9828.418)(11

−=′

+′=⎪⎩

⎪⎨⎧ ≤

=

∑ =−×=

xy

iyiyiyiyiyiz

i izDyf D

[-500,500]D [-500,500]D {420.96}D 5000 Rotated Schwefel[59]

matrix orthogonalan is M ,*M where

]10)2cos(10[)(1

212

xy

yyyfD

i ii

=

+−=∑ =π

[-5.12,5.12]D [-5.12,2]D {0}D 100 Rotated Rastrigin[59]


20)2cos1exp( )12.0exp(20)(11

213

xy

eyD

yD

yfD

i iD

i i

=

++−−−= ∑∑ ==π [-32,32]D [-32,16]D {0}D 1×10-6 Rotated

Ackley[59]


64


1)/cos(4000/1)(1 1

214

xy

iyyyfD

i

D

i ii

=

+−= ∑ ∏= = [-600,600]D [-600,200]D {0}D 1×10-6 Rotated Griewank[59]

optimum global shifted the: ],,,[

,1,_))1()(100()(

21

61

122

12

15

D

D

i iii

oooo

oxzbiasfzzzxf

=

+−=+−+−=∑ −

= +

[-100, 100]D [-100, 100]D o 490 Shifted Rosenbrock[163]

optimum global shifted the: ],,,[

,,_)10)2cos(10()(

21

912

16

D

D

i ii

oooo

oxzbiasfzzxf

=

−=++−=∑ =π

[-5, 5]D [-100, 100]D o -230 Shifted Rastrigin[163]

Table 3-4 PSO Algorithms for Comparison

PSO algorithm Parameter configurations References

GPSO ω: 0.9~0.4, c1= c2=2.0, VMAXd=0.2×Range [45] LPSO ω: 0.9~0.4, c1= c2=2.0, VMAXd=0.2×Range [54] FIPS χ=0.729, ∑ci = 4.1, VMAXd=0.5×Range [25]

HPSO-TVAC ω: 0.9~0.4, c1: 2.5~0.5, c2: 0.5~2.5, VMAXd=0.5×Range [28] DMS-PSO ω: 0.9~0.2, c1= c2=2.0, m=3, R=5, VMAXd=0.2×Range [57]

CLPSO ω: 0.9~0.4, c =1.49445, m=7, VMAXd=0.2×Range [59] OPSO ω: 0.9~0.4, c1= c2=2.0, VMAXd=0.5×Range [150]

OLPSO ω: 0.9~0.4, c =2.0, G=5, VMAXd=0.2×Range –

For a fair comparison among all the PSOs, they are tested using the same population size

of 40. Furthermore, all the algorithms use the same maximum number of function evaluations

(FEs) 2×105 in each run for each test function as suggested in [163]. Note that the number of

FEs consumed during the construction of the guidance exemplar oBesti in OLPSO are

included in this maximum FEs number allowed. Another notice is that an L32(231) OA is

suitable for all the test functions because they are all 30 dimension. For the purpose of

reducing statistical errors, each algorithm is tested 25 times independently for every function

and the mean results are used in the comparison.

3.3.3 Solution Accuracy with Orthogonal Learning Strategy

The solutions obtained by OLPSOs are compared with the ones obtained by PSOs

without OL strategy in Table 3-5. Table 3-5 compares the mean values and the standard

deviations of the solutions found. The best results are marked in boldface. The t-test results

between OLPSO-G and GPSO, and OLPSO-L and LPSO are also given, respectively.

1） Unimodal Functions

For the four unimodal functions, the results show that OLPSOs generally outperform the

traditional PSOs. For example, OLPSO-G does better than GPSO on functions f1, f2, and f3

whilst OLPSO-L outperforms LPSO on functions f1, f2, f3, and f4. The experimental results


65

show that the OL strategy brings solution with much higher accuracy to the problem. For the

very simple unimodal functions f1 and f2, OLPSO-G provides solutions with the highest

quality. However, as the problem becomes more complex, even become multimodal in high

dimension, such as the Rosenbrock’s function (f3), the performance of OLPSO-L is much

better. This is in coincidence with the general observation that a local version PSO does

better than a global version PSO on complex problems. This is because a local version PSO

draws experience from locally best particles, as opposed to the interim global best, and hence

avoids a premature convergence, although it could converge more slowly. As for the Noise

function (f4), we can observe that OLPSO-G does not show an advantage. This is perhaps

because the effect of the OL strategy is largely canceled out by the random fluctuation.

Table 3-5 Solutions Accuracy (Mean±Std) Comparisons Between PSOs With and Without OL Strategy

Function GPSO OLPSO-G t-Test LPSO OLPSO-L t-Test

f1 2.05×10-32±3.56×10-32 4.12×10-54±6.34×10-54 2.88† 3.34×10-14±5.39×10-14 1.11×10-38±1.28×10-38 3.10†

f2 1.49×10-21±3.60×10-21 9.85×10-30±1.01×10-29 2.07† 1.70×10-10±1.39×10-10 7.67×10-22±5.63×10-22 6.12†

f3 40.70±32.19 21.52±29.92 2.18† 28.08±21.79 1.26±1.40 6.14†

f4 9.32×10-3±2.39×10-3 1.16×10-2±4.10×10-3 -2.38† 2.28×10-2±5.60×10-3 1.64×10-2±3.25×10-3 4.96†

f5 2.48×103±2.97×102 3.84×102±2.17×102 28.53† 3.16×103±4.06×102 3.82×10-4±0 38.95†

f6 26.03±7.27 1.07±0.99 17.00† 35.07±6.89 0±0 25.46†

f7 1.31×10-14±2.08×10-15 7.98×10-15±2.03×10-15 8.80† 8.20×10-08±6.73×10-08 4.14×10-15±0 6.09†

f8 2.12×10-2±2.18×10-2 4.83×10-3±8.63×10-3 3.50† 1.53×10-3±4.32×10-3 0±0 1.77

f9 2.23×10-31±7.07×10-31 1.59×10-32±1.03×10-33 1.46 8.10×10-16±1.07×10-15 1.57×10-32±2.79×10-48 3.80†

f10 1.32×10-3±3.64×10-3 4.39×10-4±2.20×10-3 1.03 3.26×10-13±3.70×10-13 1.35×10-32±5.59×10-48 4.41†

f11 4.61×103±6.21×102 4.00×103±6.08×102 3.51† 4.50×103±3.97×102 3.13×103±1.24×103 5.28†

f12 60.02±15.98 46.09±12.88 3.39† 53.36±13.99 53.35±13.35 0.00

f13 1.93±0.96 7.69×10-15±1.78×10-15 10.01† 1.55±0.45 4.28×10-15±7.11×10-16 17.44†

f14 1.80×10-2±2.41×10-2 1.68×10-3±4.13×10-3 3.33† 1.68×10-3±3.47×10-3 4.19×10-8±2.06×10-7 2.42†

f15 427.93±54.98 424.75±34.80 0.24 432.33±43.41 415.95±23.96 1.65

f16 -223.18±38.58 -328.57±1.04 13.65† -234.95±18.82 -330±1.64×10-14 25.36†† The value of t with 48 degrees of freedom is significant at α=0.05 by a two-tailed test between the two algorithms.

Fig. 3-4 Convergence progresses of PSOs with and without OL strategy on unimodal functions.


66

The plots in Fig. 3-4 show the convergence progress of the mean solution values of the

25 trials during the run for functions f1 and f3. It is apparent that OLPSOs perform better than

the traditional PSOs in terms of final solution and convergence speed. It can be observed

from the figures that OLPSOs with the OL strategy converge considerably faster than the

traditional PSOs (GPSO and LPSO) without an OL strategy.

2） Multimodal Functions

As the efficiency of the OL strategy provides PSO an ability to discover, preserve, and

utilize useful information of the learning exemplars, it is expected that OLPSO can avoid

local optima and bring about improved performance on multimodal functions. Indeed, the

experimental results for functions f5 to f10 given in Table 3-5 support this intuition. OLPSO-G

surpasses GPSO on all the six multimodal functions. OLPSO-L yields the best performance

among the four PSOs on all the six multimodal functions, in terms of mean solutions and

standard deviations. In comparison, GPSO can only reach the global optimum on function f7

and f9 while LPSO on functions f7, f9, and f10. Best of all, OLPSO-L is able to find the global

optimum on all the functions and only OLPSO-L can show significantly improved

performance in reaching the global optimum 0 on the Rastrigin’s function (f6) and the

Griewank’s function (f8). These experimental results verify that the OLPSOs with the OL

strategy offer the ability of avoiding local optima to obtain the global optimum robustly in

multimodal functions.

The evolutionary progresses of the PSOs in optimizing the multimodal functions f5 and

f6 are plotted in Fig. 3-5. It can be observed that OLPSOs are able to improve solutions

steadily for a long period without being trapped in local optima. OLPSO-L appears to exhibit

the strongest search ability and can converge to the global optimum 0 in about 1.5×105 FEs

on the Rastrigin’s function. The convergent curves on the Schewefel’s function (f5) also show

that OLPSO-L has strong global search ability to avoid local optima.

Fig. 3-5 Convergence progresses of PSOs with and without OL strategy on multimodal functions.


67

3） Rotated and Shifted Functions

Functions f11 to f14 are multimodal functions with coordinate rotation while f15 and f16 are

shifted functions. In order to avoid biases of specific rotations in the tests, a new rotation is

computed before each run of the 25 independent trials according to Salomon method in [174].

Experimental results for the four rotated multimodal functions are also given in Table 3-5 and

the evolutionary progresses of f13 and f14 are plotted in Fig. 3-6. It appears that all the PSO

algorithms are affected by the coordinate rotation. However, it is interesting to observe that

the OLPSO algorithms can still reach the global optima of the rotated Ackley’s function (f13)

and the rotated Grienwank’s function (f14). All the PSOs are trapped by the rotated

Schwefel’s function (f11) and the rotated Rastrigin’s function (f12) as they become much more

difficult after coordinate rotation. However, OLPSOs still perform better than traditional

PSOs on these two problems. The experimental results also show that OLPSO-G and

OLPSO-L outperform GPSO and LPSO respectively on the two shifted function f15 and f16.

Moreover, only OLPSO-L can obtain the global optimum -330 on the shifted Rastrigin’s

function (f16). Overall, even though affected by the rotation and the shift, the comparisons

still indicate that the OL strategy is beneficial to the PSO performance, and OLPSOs

generally perform better than traditional PSOs.

Fig. 3-6 Convergence progresses of PSOs with and without OL strategy on rotated functions.

3.3.4 Convergence Speed with Orthogonal Learning Strategy

As the OL strategy can provide a promising guidance exemplar oBesti, it is natural that

OLPSO can reach more accurate solution with a faster convergence speed. In order to verify

this, more experimental results are given and compared in Table 3-6. The results given there

are the average FEs needed to reach the threshold expressed as acceptable solutions specified


68

in Table 3-3. In addition, successful rate (SR%) of the 25 independent runs for each function

are also compared. Note that the average FEs are calculated only for the runs that have been

‘successful’. As some algorithm may not succeed in reaching the acceptable solution every

run on some problems, the metric success performance (SP), defined as SP=(Average

FEs)/(SR%) [163], is also compared in Table 3-6. Table 3-6 Convergence Speed, Algorithm Reliability, and Success Performance Comparisons.

Function GPSO OLPSO-G LPSO OLPSO-L

FEs SR% SP FEs SR% SP FEs SR% SP FEs SR% SP f1 134561 100 134561 89247 100 89247 161985 100 161985 98337 100 98337 f2 141262 100 141262 101698 100 101698 171962 100 171962 114441 100 114441 f3 126343 100 126343 78749 100 78749 137934 100 137934 92233 100 92233 f4 171048 60 285080 150238 40 375595 × 0 × 186351 4 4658775f5 117710 8 1471375 40533 100 40533 × 0 × 51498 100 51498 f6 75274 100 75274 37783 100 37783 76061 100 76061 43635 100 43635 f7 152659 100 152659 109627 100 109627 189154 100 189154 126571 100 126571 f8 137576 32 429925 93336 68 137258.8 171756 80 214695 107217 100 107217 f9 128474 100 128474 80761 100 80761 153943 100 153943 90610 100 90610 f10 135620 88 154113.6 86667 96 90278.13 168060 100 168060 97534 100 97534 f11 77083 76 101425 54901 92 59675 89029 88 101169.3 54097 96 56351.04f12 100215 100 100215 66023 100 66023 107072 100 107072 68809 100 68809 f13 163356 16 1020975 111961 100 111961 × 0 × 129946 100 129946 f14 146446 32 457643.8 112053 84 133396.4 186771 68 274663.2 137850 96 143593.8f15 37203 84 44289.29 101632 96 105866.7 42935 84 51113.1 113317 100 113317 f16 4758 56 8496.429 37143 100 37143 16999 60 28331.67 43393 100 43393

Ave. SR 72.00% 92.25% 73.75% 93.50% It can be observed from the table that OLPSO-G and OLPSO-L are constantly faster

than GPSO and LPSO respectively on the tested functions. This indeed shows the advantage

of the OL strategy in constructing promising exemplar to guide the flying direction for faster

optimization speed. Moreover, with a reasonable agreement to the fact that GPSO is always

faster than LPSO, OLPSO-G is observed to be faster than OLPSO-L and is also the fastest

algorithm among the four contenders. Even the slower OLPSO-L (when compared with

OLPSO-G), still converges faster than GPSO (global version but without OL strategy) on

most of the functions. For example, in solving the Sphere function (f1), average numbers of

FEs 134561 and 161985 are needed by GPSO and LPSO respectively to reach the acceptable

accuracy 1×10-6. However, OLPSO-G uses only 89247 FEs, which indicates that it is the

fastest algorithm. OLPSO-L uses 98337 FEs to obtain the solution, which is faster not only

than LPSO, but also than GPSO.

The successful rates shown in the Table 3-6 also indicate that the OL strategy is very

promising in bringing a high reliability to PSO. The OLPSOs result in higher algorithm

reliability with 100% successful rate on most of the test functions while traditional PSOs are

sometimes trapped in the multimodal, rotated, or the shifted problems. Overall, OLPSO-L

yields the highest successful rate 93.50% averaged on all the 16 functions, and followed by

OLPSO-G, LPSO, and GPSO.


69

The experimental results have demonstrated that the OL strategy indeed can provide a

much better guidance for the particles to fly to a promising region faster. The OLPSOs with

the OL strategy are more robust and reliable in solving global optimization problems.

3.3.5 Comparisons with Other PSOs

To verigy the effectiveness and efficiency of the proposed OLPSO, the OLPSOs will be

compared with some other improved PSO variants, namely, FIPS, HPSO-TVAC, DMS-PSO,

CLPSO, and OPSO. The mean and the standard deviation (SD) of the final solutions are

given and compared in Table 3-7.

It can be observed that OLPSOs achieve the best solution on most of the functions. FIPS

performs best on the the Noise function (f4) and rotated Griewank’s function (f14). DMS-PSO

yields the best solution on the rotated Rastrigin’s function (f12). CLPSO does best on the

shifted Rosenbrock function (f15) and obtains the same best mean solution as OLPSO-L does

on the Schewefel’s function (f5) and the shifted Rastrigin function (f16). OLPSO-G performs

best on f1, and f2.Overall, OLPSO-L performs best on f3, f5, f6, f7, f8, f9, f10, f11, f13, and f16, i.e.,

10 out of the 16 functions.

On the unimodal functions, OLPSO-G is shown to offer superior performance among all

the PSOs. On the multimodal functions, OLPSO-L generally outperform all the other PSO

variants. On the coordinate rotated and shifted functions, OLPSOs also generally do better

than other PSOs. OLPSO-G can still obtain the global optimum of the rotated Ackley

function (f13) while OLPSO-L can still obtain the global optima of the rotated Ackley (f13)

and the rotated Griewank (f14) functions. Same as other PSOs, the OLPSO algorithms failed

on the rotated Schwefel (f11) and Rastrigin (f12) functions, as they become much harder after

rotation. However, OLPSO-L is still the best algorithm on f11 and the results are comparable

with DMS-PSO on f12. Only can FIPS, DMS-PSO, OPSO, and our OLPSOs achieve the

global optimum on f13, only can FIPS and OLPSO-L achieve the global optimum on f14, and

only CLPSO and OLPSO-L achieve the global optimum on f16.

Table 3-7 also ranks the algorithms on performance in terms of the mean solution

accuracy. It can be observed from the final rank that OLPSO-L offers the best overall

performance, while OLPSO-G is the second best, followed by CLPSO, FIPS, HPSO-TVAC,

DMS-PSO, OPSO, GPSO, and LPSO. Both FIPS and HPSO-TVAC ranks the fourth.


70

Table 3-7 Search Result Comparisons of PSOs on 16 Global Optimization Functions

Function GPSO LPSO FIPS HPSO-TVAC DMS-PSO CLPSO OPSO OLPSO-G OLPSO-L

f1 Mean 2.05×10-32 3.34×10-14 2.42×10-13 2.83×10-33 2.65×10-31 1.58×10-12 6.45×10-18 4.12×10-54 1.11×10-38

SD 3.56×10-32 5.39×10-14 1.73×10-13 3.19×10-33 6.25×10-31 7.70×10-13 4.64×10-18 6.34×10-54 1.28×10-38

Rank 4 7 8 3 5 9 6 1 2

f2 Mean 1.49×10-21 1.70×10-10 2.76×10-8 9.03×10-20 1.57×10-18 2.51×10-8 1.26×10-10 9.85×10-30 7.67×10-22

SD 3.60×10-21 1.39×10-10 9.04×10-9 9.58×10-20 3.79×10-18 5.84×10-9 5.58×10-11 1.01×10-29 5.63×10-22

Rank 3 7 9 4 5 8 6 1 2

f3 Mean 40.70 28.08 25.12 23.91 41.58 11.36 49.61 21.52 1.26

SD 32.19 21.79 0.51 26.51 30.25 9.85 36.54 29.92 1.40 Rank 7 6 5 4 8 2 9 3 1

f4 Mean 9.32×10-3 2.28×10-2 4.24×10-3 9.82×10-2 1.45×10-2 5.85×10-3 5.50×10-2 1.16×10-2 1.64×10-2

SD 2.39×10-3 5.60×10-3 1.28×10-3 3.26×10-2 5.05×10-3 1.11×10-3 1.70×10-3 4.10×10-3 3.25×10-3 Rank 3 7 1 9 5 2 8 4 6

f5 Mean 2.48×103 3.16×103 9.93×102 1.59×103 3.21×103 3.82×10-4 2.93×103 3.84×102 3.82×10-4

SD 2.97×102 4.06×102 5.09×102 3.26×102 6.51×102 1.28×10-07 5.57×102 2.17×102 0 Rank 6 8 4 5 9 2 7 3 1

f6 Mean 26.03 35.07 65.10 9.43 27.15 9.09×10-5 6.97 1.07 0

SD 7.27 6.89 13.39 3.48 6.02 1.25×10-4 3.07 0.99 0 Rank 6 8 9 5 7 2 4 3 1

f7 Mean 1.31×10-14 8.20×10-8 2.33×10-7 7.29×10-14 1.84×10-14 3.66×10-7 6.23×10-9 7.98×10-15 4.14×10-15

SD 2.08×10-15 6.73×10-8 7.19×10-8 3.00×10-14 4.35×10-15 7.57×10-8 1.87×10-9 2.03×10-15 0 Rank 3 7 8 5 4 9 6 2 1

f8 Mean 2.12×10-2 1.53×10-3 9.01×10-12 9.75×10-3 6.21×10-3 9.02×10-9 2.29×10-3 4.83×10-3 0

SD 2.18×10-2 4.32×10-3 1.84×10-11 8.33×10-3 8.14×10-3 8.57×10-9 5.48×10-3 8.63×10-3 0 Rank 9 4 2 8 7 3 5 6 1

f9 Mean 2.23×10-31 8.10×10-16 1.96×10-15 2.71×10-29 2.51×10-30 6.45×10-14 1.56×10-19 1.59×10-32 1.57×10-32

SD 7.07×10-31 1.07×10-15 1.11×10-15 1.88×10-29 1.02×10-29 3.70×10-14 1.67×10-19 1.03×10-33 2.79×10-48

Rank 3 7 8 5 4 9 6 2 1

f10 Mean 1.32×10-3 3.26×10-13 2.70×10-14 2.79×10-28 2.64×10-3 1.25×10-12 1.46×10-18 4.39×10-4 1.35×10-32

SD 3.64×10-3 3.70×10-13 1.57×10-14 2.18×10-28 4.79×10-3 9.45×10-13 1.33×10-18 2.20×10-3 5.59×10-48

Rank 8 5 4 2 9 6 3 7 1

f11 Mean 4.61×103 4.50×103 4.41×103 5.32×103 4.04×103 4.39×103 4.48×103 4.00×103 3.13×103

SD 6.21×102 3.97×102 9.94×102 7.00×102 5.68×102 3.51×102 1.03×103 6.08×102 1.24×103 Rank 8 7 5 9 3 4 6 2 1

f12 Mean 60.02 53.36 1.50×102 52.90 41.97 87.14 63.78 46.09 53.35 SD 15.98 13.99 14.48 12.54 9.74 10.76 19.73 12.88 13.35

Rank 6 5 9 3 1 8 7 2 4

f13 Mean 1.93 1.55 3.16×10-7 9.29 2.42×10-14 5.91×10-5 1.49×10-8 7.69×10-15 4.28×10-15

SD 0.96 0.45 1.00×10-7 2.07 1.52×10-14 6.46×10-5 6.36×10-9 1.78×10-15 7.11×10-16

Rank 8 7 5 9 3 6 4 2 1

f14 Mean 1.80×10-2 1.68×10-3 1.28×10-8 9.26×10-3 1.02×10-2 7.96×10-5 1.28×10-3 1.68×10-3 4.19×10-8 SD 2.41×10-2 3.47×10-3 4.29×10-8 8.80×10-3 1.24×10-2 7.66×10-5 3.70×10-3 4.13×10-3 2.06×10-7

Rank 9 5 1 7 8 3 4 6 2

f15 Mean 427.93 432.33 424.83 494.20 502.51 403.07 2.45×107 424.75 415.94 SD 54.98 43.41 25.37 96.54 95.18 13.50 4.40×107 34.80 23.96

Rank 5 6 4 7 8 1 9 3 2

f16 Mean -223.18 -234.95 -245.77 -318.33 -303.17 -330 -284.11 -328.57 -330 SD 38.58 18.82 22.08 5.75 5.01 3.39×10-5 13.62 1.04 1.64×10-14

Rank 9 8 7 4 5 2 6 3 1 Total rank 97 104 89 89 91 76 96 50 28 Ave. rank 6.06 6.50 5.56 5.56 5.69 4.75 6.00 3.13 1.75 Final rank 8 9 4 4 6 3 7 2 1 Algorithms GPSO LPSO FIPS HPSO-TVAC DMS-PSO CLPSO OPSO OLPSO-G OLPSO-L


71

Table 3-8 Convergence Speed, Algorithm Reliability, and Success Performance Comparisons Among

Different PSO Variants

Function GPSO LPSO FIPS HPSO-TVAC DMS-PSO CLPSO OPSO OLPSO-G OLPSO-L

f1 FEs 134561 161985 118306 63982 138103 139554 92908 89247 98337

SR% 100 100 100 100 100 100 100 100 100 SP 134561 161985 118306 63982 138103 139554 92908 89247 98337

f2 FEs 141262 171962 165502 78944 147644 172886 134640 101698 114441

SR% 100 100 100 100 100 100 100 100 100 SP 141262 171962 165502 78944 147644 172886 134640 101698 114441

f3 FEs 126343 137934 48456 50628 125573 108669 71654 78749 92233

SR% 100 100 100 100 100 100 100 100 100 SP 126343 137934 48456 50628 125573 108669 71654 78749 92233

f4 FEs 171048 × 91081 × 194220 133550 × 150238 186351

SR% 60 0 100 0 24 100 0 40 4 SP 285080 × 91081 × 809250 133550 × 375595 4658775

f5 FEs 117710 × 133646 56683 104422 65429 43200 40533 51498

SR% 8 0 100 92 4 100 4 100 100 SP 1471375 × 139214.6 61611.96 2610550 65429 1080000 40533 51498

f6 FEs 75274 76061 79421 6096 74803 44000 24768 37783 43635

SR% 100 100 100 100 100 100 100 100 100 SP 75274 76061 79421 6096 74803 44000 24768 37783 43635

f7 FEs 152659 189154 183341 102496 162400 190767 155088 109627 126571

SR% 100 100 100 100 100 100 100 100 100 SP 152659 189154 183341 102496 162400 190767 155088 109627 126571

f8 FEs 137576 171756 133787 66965 141489 167486 110232 93336 107217

SR% 32 80 100 28 56 100 80 68 100 SP 429925 214695 133787 239160.7 252658.9 167486 137790 137258.8 107217

f9 FEs 128474 153943 94368 74033 137909 124779 77587 80761 90610

SR% 100 100 100 100 100 100 100 100 100 SP 128474 153943 94368 74033 137909 124779 77587 80761 90610

f10 FEs 135620 168060 107315 75483 145063 138209 86716 86667 97534

SR% 88 100 100 100 76 100 100 96 100 SP 154113.6 168060 107315 75483 190872.4 138209 86716 90278.13 97534

f11 FEs 77083 89029 115196 66394 109220 128544 31920 54901 54097

SR% 76 88 68 28 96 100 60 92 96 SP 101425 101169.3 169405.9 237121.4 113770.8 128544 53200 59675 56351.04

f12 FEs 100215 107072 × 8208 88935 146299 107942 66023 68809

SR% 100 100 0 100 100 92 100 100 100 SP 100215 107072 × 8208 88935 159020.7 107942 66023 68809

f13 FEs 163356 × 187032 × 169314 × 161856 111961 129946

SR% 16 0 100 0 100 0 100 100 100 SP 1020975 × 187032 × 169314 × 161856 111961 129946

f14 FEs 146446 186771 150433 105910 163996 × 161083 112053 137850

SR% 32 68 100 32 36 0 88 84 96 SP 457643.8 274663.2 150433 330968.8 455544.4 × 183048.9 133396.4 143593.8

f15 FEs 37203 42935 75137 129660 140749 129159 75960 101632 113317

SR% 84 84 92 48 56 100 24 96 100 SP 44289.29 51113.1 81670.65 270125 251337.5 129159 316500 105866.7 113317

f16 FEs 4758 16999 98131 27875 57607 39619 25459 37143 43393

SR% 56 60 68 100 100 100 100 100 100 SP 8496.429 28331.67 144310.3 27875 57607 39619 25459 37143 43393

Ave. SR 72.00% 73.75% 89.00% 70.50% 78.00% 87.00% 78.50% 92.25% 93.50% SR rank 8 7 3 9 6 4 5 2 1

Algorithms GPSO LPSO FIPS HPSO-TVAC DMS-PSO CLPSO OPSO OLPSO-G OLPSO-L

In order to compare the convergence speed, algorithm reliability, and success

performance, Table 3-8 gives the mean FEs to reach the acceptable accuracy among the


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success runs, the successful rate, and the success performance. The results show that FIPS

and HPSO-TVAC converges very fast on most of the function. Especially on unimodal

functions, HPSO-TVAC is fastest on f1 and f2, and FIPS is fastest on f3 and f4. In addition,

HPSO-TVAC converges fast on some multimodal functions, but the rapid convergence rate is

not able to ensure that HPSO-TVAC can achive the global optimum solution. For example,

HPSO-TVAC has low successful rate on some multimodal functions, which limits the real

value of the algorithm.

However, OLPSOs do much better in reaching the global optima robustly, as measured

by the successful rate. Even though OLPSOs are sometimes slower than HPSO-TVAC, they

are still general faster than many other PSOs.

Moreover, OLPSOs generally outperform the contenders with higher successful rate.

One interesting fact, is the “total failure” on some functions of some algorithms, that is to say,

the algorithms can not obtain an accepted solution in all 25 runs. For example, the successful

rate of LPSO on f4, f5 and f13 are 0, FIPS fails on f4 and f13, and CLPSO cannot achive

accepted solutions on f13 and f14 while OPSO on f4. In contrast, OLPSO-G and OLPSO-L

successfully obtain accepted solutions on all functions. OLPSO-G gets a successful rate of

100% on f1, f2, f3, f5, f6, f7, f9, f12, f13, and f16, total 10 functions, while OLPSO-L also has a

successful rate of 100% on 16 functions, f1, f2, f3, f5, f6, f7, f8, f9, f10, f12, f13, f15, and f16. In a

conclusion, OLPSO-L has the highest average successful rate of 93.50%, OLPSO-G has the

second highest one of 92.25%, followed by FIPS, CLPSO, OPSO, DMS-PSO, LPSO, GPSO,

and HPSO-TVAC.

Overall, from the great performance of OLPSO on convergence speed, algorithm

reliability, and success performance, we can see that OLPSO is an effective and efficient

global optimization algorithm.

3.3.6 Comparisons with Other Evolutionary Algorithms

The proposed OLPSOs are further compared with some state of the art evolutionary

algorithms (EAs) in Table 3-9. These algorithms include variants of EAs, such as fast

evolutionary programming (FEP) with Cauchy mutation (1999) [162], orthogonal GA with

quantization (OGA/Q) (2001) [167], estimation of distribution algorithm with local search

(EDA/L) (2004) [175], evolution strategy with covariance matrix adaptation (CMA-ES)

(2005) [176], and adaptive differential evolution (JADE) with optional external archive (2009)

[177]. As OLPSO-L generally outperforms OLPSO-G in global optimization, we only


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compare OLPSO-L with these algorithms in Table 3-9. The results of the compared

algorithms are all derived directly from their corresponding references except that the results

of CMA-ES are obtained by our independent experiments on these functions based on the

provided source code [176].

OGA/Q is an EA with orthogonal initialization and orthogonal crossover. It yielded

good performance and did best on 5 of the 10 functions, indicating the advantages of the

OED method. Our OLPSO-L also does best on 5 of the 10 functions. Specifically, OGA/Q

seems to be better than OLPSO-L on unimodal functions (e.g., f1 and f2) while OLPSO-L

does better than OGA/Q on some of the multimodal functions (e.g., f9 and f10). CMA-ES does

best on the Rosenbrock function (f3) and JADE did best on the Noise function (f4). The results

show that OLPSO-L has very competitive performance when compared with these state of

the art EAs, especially for its strong global search ability on multimodal functions. OLPSO-L

works best on f5, f9, and f10, the same best with OGA/Q, EDA/L, and JADE on f6, the same

best with OGA/Q and EDA/L on f8, and the second best on f7. Table 3-9 Result Comparisons of OLPSO-L and Some State of the Art Evolutionary Computation

Algorithms With The Existing Results Reported in The Corresponding References

Func FEP [162] OGA/Q [167] EDA/L [175]† CMA-ES [176]‡ JADE [177] OLPSO-L f1 5.7×10-4±1.3×10-4 0±0 N/A 4.54×10-16±1.13×10-16 1.3×10-54±9.2×10-54 1.11×10-38±1.28×10-38

f2 8.1×10-3±7.7×10-4 0±0 N/A 2.32×10-3±9.51×10-3 3.9×10-22±2.7×10-21 7.67×10-22±5.63×10-22

f3 5.06±5.87 0.75±0.11 4.324×10-3 2.33×10-15±7.73×10-16 0.32±1.1 1.26±1.40 f4 7.6×10-3±2.6×10-3 6.30×10-3±4.07×10-4 N/A 5.92×10-2±1.73×10-2 6.8×10-4±2.5×10-4 1.64×10-2±3.25×10-3

f5 14.98±52.6♠ 3.03×10-2±6.447×10-4♠ 2.9×10-3♠ 3.15×103±5.79×102 7.1±28 3.82×10-4±0 f6 4.6×10-2±1.2×10-2 0±0 0 1.76×102±13.89 0±0 0±0 f7 1.8×10-2±2.1×10-3 4.440×10-16±3.989×10-17 4.141×10-15 12.12±9.28 4.4×10-15±0 4.14×10-15±0 f8 1.6×10-2±2.2×10-2 0±0 0±0 9.59×10-16±3.51×10-16 2.0×10-4±1.4×10-3 0±0 f9 9.2×10-6±3.6×10-6 6.019×10-6±1.159×10-6 3.654×10-21 1.63×10-15±4.93×10-16 1.6×10-32±5.5×10-48 1.57×10-32±2.79×10-48

f10 1.6×10-4±7.3×10-5 1.869×10-4±2.615×10-5 3.485×10-21 1.71×10-15±3.70×10-16 1.4×10-32±1.1×10-47 1.35×10-32±5.59×10-48

†The standard deviation is not available in [175] and N/A means the results are nor available. ‡The results of CMA-ES are obtained by our independent experiments on these functions.

♠The mean value of f5 has been added to 418.9829×D to make the global optimal value is equal to 0.

3.3.7 Parameter Analysis

In order to investigate the influence of G on the performance of the OLPSO algorithm,

empirical studies are carried out on relevant functions, namely the Sphere, Rosenbrock,

Schwefel, Rastrigrin, Ackley, and Grienwank functions listed in Table 3-3 as the f1, f3, f5, f6,

f7, and f8, respectively. Parameter G controls the update frequency of the guidance vector

oBesti. As discussed in the previous subsection, the particle will use vector oBesti as the

learning exemplar steadily and reconstruct the oBesti only after a stagnation of pBesti for G


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generations. As can be imagined, if G is too small, the particles will reconstruct the guidance

exemplar oBesti frequently. This may waste computations on OED when it is not indeed

necessary. Also, the search direction will not be steady if oBesti changes frequently. On the

other hand, if G is too large, the particles will waste much computation on the local optima

with an oBesti which is not effective any longer. In order to further analyze the effect of G,

different values for G from 0 to 10 are tested, and two OLPSO versions that based on a global

topology (OLPSO-G) and a local topology (OLPSO-L) are simulated. The results of the

investigation are shown in Fig. 3-7(a) and Fig. 3-7(b) with averagely 25 independent runs for

the OLPSO-G and the OLPSO-L, respectively. The figures reveal that a value of G around 5

offers the best performance. This also indicates OLPSO indeed benefits from the OL strategy

by the steadily guidance of a promising learning exemplar. Therefore, a reconstruction gap of

G=5 is used in this chapter.

Fig. 3-7 OLPSO performance with different values of G. (a) OLPSO-G. (b) OLPSO-L.

3.3.8 Discussions

Experimental results and comparisons verify that the OL strategy indeed helps the

OLPSOs perform better than the traditional PSOs and most existing improved PSO variants

on most of the test functions, in terms of solution accuracy, convergence speed, and

algorithm reliability. OLPSOs offer not only better performance in global optimization, but

also finer-grain search ability, owing to the OL strategy that can discover, preserve, and

utilize useful information from the search experiences.

The OPSO in [150]also uses the OED method to improve the algorithm performance.

The particle in OPSO uses OED on both the cognitive learning and social learning

components to construct the position for the next move. The particle velocity is obtained by

calculating the difference between the new position and the current position. Differently, our


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proposed OLPSO emphasizes the learning strategy and uses OED to design an OL strategy.

The OL strategy uses OED to construct a promising and efficient exemplar to guide the

particle’s flying. OLPSO works under the framework of traditional PSO except that particle

in OLPSO learns from its constructed guidance exemplar oBesti instead of pBesti and nBesti,

i.e., uses Eq. (3-4) instead of Eq. (3-5). Therefore, the useful information in pBesti and nBesti

can be discovered and preserved through the OL strategy.

OLPSO benefits from the following three advantages. First, since only one learning

exemplar oBesti is used, the guidance would be more steady and can weaken the ‘oscillation’

phenomenon. Second, as oBesti is constructed via OED on pBesti and nBesti, the useful

information can be discovered and preserved to predict promising region for guiding the

particle, weakening the ‘two steps forward, one step back’ phenomenon. Third, as the oBesti

is used as the learning exemplar steadily until it can not improve the particle’s fitness for G

generations, it can guide the particle to fly towards the promising region steadily, resulting in

better global search performance. The experimental results and comparisons support these

advantages.

The comparisons between OLPSO-G and OLPSO-L show that for unimodal functions

OLPSO-G outperforms OLPSO-L on both accuracy and speed, whilst for multimodal

functions OLPSO-L is better for final solution accuracy. This may be due to that OLPSO-L is

based on the local version PSO that provides a better diversity and avoids premature

convergence. Nevertheless, OLPSO-L can also do very well on unimodal functions and it

outperforms most of the existing PSOs. Hence, OLPSO-L is the recommended global

optimizer here. Moreover, the comparisons with some state of the art EAs show that

OLPSO-L is general better than, or at least comparable to these variants of EAs.

3.4 Chapter Summary

In this chapter, we introduced the orthogonal experiment design and orthogonal

prediction technique in machine learning into PSO and presented a new orthogonal learning

particle swarm optimization by designing an OL strategy to discover useful information from

a particle’s personal best position pBesti and its neighborhood’s best position nBesti.

Comprehensive experimental tests have been conducted on 16 benchmarks including

unimodal, multimodal, coordinate-rotated, and shifted functions. The experimental results

demonstrate the high effectiveness and the high efficiency of the OL strategy and the OLPSO

algorithms. The resultant OLPSO-G and OLPSO-L algorithms both significantly outperform


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other existing PSO algorithms on most of the functions tested, contributing to higher solution

accuracy, faster convergence speed, and stronger algorithm reliability. Comparisons are also

made with some state of the art EAs, and the OLPSO algorithm shows very promising

performance.

The features and advantages of the proposed OL strategy and OLPSO algorithms are:

1) Only one guidance vector is used in OL strategy to guide the fly of the particles,

which is able to avoid the ‘oscillation’ phenomenon in traditional PSO due to the guidance of

two vector, obtain sustained and stable guidance information, and ensure the convergence of

the algorithm.

2) As a orthogonal prediction technique based on OED, this new OL strategy helps a

particle discover useful information from a particle’s personal best position pBesti and its

neighborhood’s best position nBesti and construct a more promising and efficient guidance

exemplar oBesti to adjust its flying velocity and direction, which results in easing the ‘two

steps forward, one step back’ phenomenon, offering rapid and correct direction guidance to

the particles, and accelerating the convergence to global optimum solution.

3) OL is an operator and can be applied to PSO with any topological structures, such as

the star (global version), the ring (local version), the wheel, and the von Neumann structures.

Without loss of generality, we applied it is to both the global and the local versions of PSO,

yielding the novel OLPSO-G and OLPSO-L algorithms to verify that the OL strategy is able

to discover useful information, and use it to enhance the solution accuracy, convergence rate,

and algorithm reliability with orthogonal technique.

4) OL follows the same simple algorithm framework of PSO, and introduces only one

parameter G to control the update frequency of guadiance vector, which makes the algorithm

still easy to understand and implement. Thus, OL strategy and OLPSO algorithm not only

retains the simplicity of the traditional PSO, but also greatly improves the performance of the

algorithm.

In a conclusion, this chapter introduced the orthogonal experiment design and orthogonal

prediction technique into PSO, and designed an easy understand and implement OL learning

strategy and OLPSO algorithm. The OL strategy plays a great effect on the rapid global

convergence of PSO algorithm, and improves the performance of the algorithm on complex

multimodal optimization problems. OL and OLPSO is an important and successful

exploration to the machine learning aided PSO algorithms.

Chapter 4 Multiple Populations for Multiple Objectives: A Co-evolutionary Technique for Solving Multi-objective Optimization Problems based on Ensemble Learning Techique in Machine Learning

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Chapter 4 Multiple Populations for Multiple Objectives: A

Co-evolutionary Technique for Solving Multi-objective

Optimization Problems based on Ensemble Learning Techique in

Machine Learning

4.1 Introduction

Particle swarm optimization (PSO) has been widely applied on many optimziaiton areas

due to its simple algorithm procedure and effective performance. An active research trend in

PSO is to extend PSO to multi-objective optimization problem (MOP) [178].

Multi-objective optimization problems (MOP) have received considerable attentions

over the past several decades because of its significance in a large number of real-world

applications [179][180]. An MOP has multiple objectives that often contradict with each

other, so the optima of an MOP are a group of solutions but not just a single point. Without

any preference of the decision maker for the objectives, each one in the solutions group is an

optimum solution. For example, solution A is better than solution B on the first objective,

while solution B is better than solution A on the second objective, so these two solutions have

no significant difference. Thus, the algorithms addressing the MOP should achive a group of

solutions covering the entire Pareto front (PF). As the population-based characteristic of

evolutionary computation (EC) algorithms meets the requirement of a set of solutions of

MOP, many researchers try to extend the EC algorithms to MOP [181][182][183]. However,

when using multi-objective evolutionary algorithm (MOEA) to solve MOP, a problem is that

how to select good individuals into the next generation. Since an MOP has multiple

objectives that often contradict with each other, it is difficult to say whether one individual is

better than another if it is better on one objective but is worse on another objective. This is

the fitness assignment problem encountered by the MOEA researchers. As the EC algorithms

derive from the concept of “survival of the fittest” in Darwin’s natural selection law, it would

cause the search inefficiency if we can not address the fitness assignment problem. Therefore,

one of the most significant research topics in MOEA is to design a suitable method to assign

an individual’s fitness [184][185].


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To overcome the fitness assignment difficulty in MOEA, various techniques like the

objectives aggregation technique, the objectives alternate technique, and the Pareto based

technique have been proposed in the literature. The objective aggregation technique sums the

objectives to a single objective by weights and then optimizes the formed single objective.

However, this technique requires the users to determine the weights for different objectives.

Moreover, only one solution can be obtained in a run by using this technique. The objectives

alternate technique sorts the objectives according to their importance and optimizes them

alternately. However, the ordering of the objectives may affect the performance significantly

while determining the importance of different objectives is often problem dependent. The

third popular technique is to apply the Pareto dominance to rank the individuals and assign

fitness to them. The domination rank technique may be helpful for approximating the Pareto

front (PF). However, as the Pareto dominance is a partial order relation, it is difficult to select

individuals for the next generation. Thus the obtained solutions may still not spread along the

whole PF if the selection operator can not maintain sufficient diversity. Therefore, developing

MOEA that can assign the individual’s fitness easily and also can keep the diversity to

approximate the whole true PF is still a challenging research topic in the MOEA community.

According to the research on the published papers, this chapter find that most

multiobjective evolutionary algorithm (MOEA) and multi-objective PSO (MOPSO) adopts

the algorithm paragiam of single population for multi-objective, hence each individual in the

population must track all objectives, which result in the difficulty of the fitness assignment

problem mentioned above. As there are multiple objectives in the MOP, can we use multiple

populations, instead of only one population, to optimize the MOP?

Enemble learning (EL) is a popular technique in machine learning area. In machine

learning, the researchers find that using only one classifier is difficult to make an efficient

classification for the tranning data with various type. The main idea of EL is to use multiple

classifiers to improve learning efficiency [186]. The research work in machine learning has

reported that EL has higher efficiency [187] compared with single classifier method. Inspired

by the idea of EL, this chapter proposed a multiple populations for multiple objectives

(MPMO) framework. The motivations of MPMO lie in that: as there are multiple objectives

in the MOP, can we use multiple populations, instead of only one population, to optimize the

MOP? Since it is difficult to consider all the objectives as a whole in one population, can we

treat them separately in different populations? Combine the idea of ensemble learning in

machine learning with the design of MOEA and MOPSO, a novel technique termed multiple


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populations for multiple objectives (MPMO) is proposed in this chapter.Therefore, the idea of

the MPMO technique is novel and is different from the above techniques: instead of tackling

all the objectives together as a whole by the same population, MPMO uses multiple

populations to deal with the multiple objectives, with each population being corresponded

with only one objective and all populations cooperating to approximate the whole PF.

Fig. 4-1 illustrates the framework of the MOP algorithm with M populations based on

the MPMO technique to solve an MOP with M objectives. In every generation, the

individuals in each population calculate all the objective functions like that in traditional

MOP algorithms. However, when executing evolutionary operators like selection, the fitness

value of an individual in the mth population is assigned by the mth objective function of the

MOP, where 1≤m≤M. This way, the individuals would not be confused by different

conflicting objectives any more, but are guided by the corresponding objective to search

different regions of the PF. However, as each population focuses on optimizing one objective

only, it may cause the problem that MPMO leads the individuals in each population to the

margin of the corresponding objective, resulting in inefficient approximation of the whole PF.

Therefore, another feature of MPMO is that it requires the algorithm to design an information

sharing strategy, as shown in Fig. 4-1. This way, different populations can share their search

information and communicate with others through the information sharing strategy to

approximate the whole PF efficiently.

Fig. 4-1 Framework of MPMO based algorithm for solving MOP.

Therefore, the proposed MPMO inspred by the ensemble learning is a basis for the

design of co-evolutionary multiple population algorithm for multiple objectives. During the

last two decades, the co-evolutionary concept has also been used by scientists from the EC

community [188][189][190]. However, multiple populations for multiple objectives based on

co-evolutionary has been rarely reported. Being a general technique, it is straightforward to


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implement MPMO algorithm which can accommodate any existing single objective

optimization algorithm in each population. In this chapter, by considering that particle swarm

optimization (PSO) is a simple yet powerful global optimizer with very fast convergence

speed, we adopt PSO for each swarm and design a co-evolutionary multi-swarm PSO

(CMPSO) as an instantiation of MPMO to solve MOP.

Based on the MPMO framework, CMPSO uses an external shared archive to implement

the information sharing strategy, with two novel designs being developed to enhance the

algorithm performance. The first design is to modify the particle velocity updating equation

with information obtained from externally shared archive. The shared archive is used to store

the non-dominated solutions found by different swarms and is updated every generation. The

velocity and position of a particle are not only updated by considering its personal experience

and its swarm’s experience, but also the experience fetched from the archive. Therefore, all

the swarms can share their search information thoroughly via the shared archive. This is

useful for the algorithm to accelerate the approximation the whole PF. The second design is

to utilize an elitist learning strategy (ELS) to update the archive in order to introduce

sufficient diversity so as to avoid the occurrence of local PFs. This may be helpful for MOPs

with multimodal objective functions or with complicated Pareto sets.

Inspired by the ensemble learning technique in machine learning, this chapter designed a

framework termed multiple populations for multiple objectives and proposed a

co-evolutionary multi-swarm PSO termed CMPSO. The innovations and advantages of the

CMPSO algorithm are as follows.

1) Different from existing algorithms that treat an MOP as a whole by considering all

the objectives together in a population, a framework termed multiple population for multiple

objectives (MPMO) is designed based on the idea of ensemble learning. With the MPMO

framework, each swarm is optimized by taking only one objective into account. Then,

different swarms will cooperate with each other to approximate the whole PF efficiently.

2) MPMO is a general technique applicable to many evolutionary algorithms. This

chapter adopts PSO for each population, hence proposed a co-evolutionary multi-swarm PSO

(CMPSO) and designed an external shared archive to implement the information sharing and

cevolution strategy among population.

3) CMPSO developes two novel designs, modifying particle velocity updating equation

to accelerate the convergence of the algorithm and using an elist learning strategy (ELS) in

the archive update process to avoid local optima.


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4.2 Multi-objective Optimization Problem

4.2.1 Related concept of MOP

A minimization MOP can be described as follow:

Minimize F(X) = {F1(X), F2(X), …, FM(X)} (4-1)

where the X = [X1, X2, …, XD] ∈ ℜD is a point in the D dimensional decision space (search

space) and the F = [F1, F2, …, FM] ∈ ΩM is the objective space with M minimization

objectives. For minimization MOP as (3), some definitions of MOP are given as follows：

Definition 1: Pareto domination. Given two vectors U = [U1, U2, …, UM] ∈ ΩM and

W = [W1, W2, …, WM] ∈ ΩM in the objective space. We say that U dominates (is better

than) W if Um≤Wm, for all m=1, 2, …, M, and U≠W.

Definition 2: Pareto optimal. Given a vector X = [X1, X2, …, XD] ∈ ℜD in the decision

space. We say that X is Pareto optimal if there is no X* ∈ ℜD such that F(X*) dominates

F(X).

Definition 3: Pareto set. The Pareto set PS is defined as

PS = {X ∈ ℜD and X is Pareto optimal} (4-2)

Definition 4: Pareto front. The Pareto front PF is defined as：

PF = {F(X) | X ∈ PS} (4-3)

4.2.2 Related work on MOP

This section will review the related work on MOP, and give a description from the

design and application of MOEA and MOPSO.

There are many algorithms, like MOEAs and MOPSOs, for solving MOP. Some

researchers used aggregation approach to solve MOP [191]. That is, the multiple objectives

are weighted and summed to form a single objective and the obtained single objective

problem is then optimized. However, the weights for different objectives are dependent on

the problem or the decision makers. Therefore they are difficult to be determined. Moreover,

the aggregation approach obtains only one solution in a run which is not sufficient in

practical applications. To overcome these disadvantages, Zhang and Li [185] proposed to

decompose the MOP into different scalar optimization sub-problems and use different

weights to these sub-problems to obtain a set of Pareto solutions in a single run. Some other


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researchers proposed to optimize the objectives alternately when solving MOP [56]. However,

as the approach used in [56] involves using two objectives for determining the neighbors and

the neighborhood best particle respectively, it seems to be useful only in MOP with two

objectives. Moreover, determining the importance of different objectives is problem

dependent and the ordering of the objectives may affect the performance significantly.

Other researchers applied the concept of Pareto dominance to solve MOP. The

multi-objective genetic algorithm (GA) assigns a rank to each individual according to the

number of other individuals that dominate it [192]. Then the fitness is assigned to the

individual based on its rank. The niching Pareto GA compares every two individuals based

on the Pareto domination tournament strategy [193]. The non-dominated sorting genetic

algorithm (NSGA-II) sorts all the individuals according to the Pareto domination relationship

and selects individuals with better ranks to form the next generation population [184].

Moreover, many MOPSOs adopt the Pareto domination concept when assigning the fitness

value of the individuals [29][194], e.g., when determining the personal historically best

position [29] or selecting the globally best position [194].

Recently, some new work has been reported to use new techniques to help solve MOP

more efficiently. In the studies of Rachmawati and Srinivasan [195], Karahan and Koksalan

[196], and Zitzler et al. [197], the preference information is used in MOEAs for better

selecting the individuals to the next generation. In designing efficient MOEAs, Wang et al.

[198] used hybrid technique, Adra et al. [199] used a convergence acceleration operator,

Avigad and Moshaiov [200] used interactive concept, Lara et al. [201] used a hill climber

with sidestep local search strategy, Song and Kusiak [202] used a data mining process, and

Zhang et al. [203] used a Gaussian process model. Moreover, some multi-objective

optimization algorithms based on memetic algorithm (MA) [204], quantum GA [205],

differential evolution (DE) [206][207], and estimation of distribution algorithm (EDA) [208]

have also been proposed in recent years.

4.3 CMPSO for MOP

4.3.1 CMPSO Evolutionary Process

Suppose that there are M objectives in the MOP, and therefore there are M swarms

working concurrently in CMPSO to optimize the MOP. The evolutionary process in each

swarm is similar to that in a conventional PSO which is used to optimize a single objective


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problem. Without loss of generality, we herein consider only one of the M swarms, indicating

by index of m, to describe the evolutionary process.

In the initialization, each particle i in the mth swarm randomly initializes its velocity Vim

and position Xim, evaluates Xi

m, and lets pBestim be the same as Xi

m. In every generation

during the evolutionary process, for the mth swarm, each particle i updates its velocity and

position by the influences of its personal historically best position pBestim, the swarm’s

globally best position gBestm, and an archive position Archiveim =

1 2, , ,

i i iD

m m mA A A⎡ ⎤⎣ ⎦ selected

from the archive by the particle i. The velocity update is as

)()()( 332211mid

midd

mid

mdd

mid

midd

mid

mid XArcXGrcXPrcVV −+−+−+= ω (4-4)

where d is the index of the dimension and r3d is a random value in the range of [0, 1]. The

position update is as m

idmid

mid VXX += (4-5)

In the velocity update equation, the term )(33mid

midd XArc − is the sharing information

from the shared archive. With the help of solution information in the shared archive, the

particle can use the search information not only from its own swarm, but also from other

swarms. The particle is expected to search along the whole PF by using the whole search

information of all the swarms instead of being attracted to the margin only by the search

information of its own swarm. Therefore the algorithm can approximate the whole PF fast

with the help of the archived information. The Archiveim is chosen by randomly selecting a

solution from the shared archive by the particle i. A random selection method is rapid, has

advantages of high diversity, and is with low computational cost. Therefore it is adopted in

this chapter. One notice is that if the archive is empty then the Archiveim is selected randomly

from the gBests of the other M–1 swarms excluding the mth swarm itself.

4.3.2 CMPSO Archive Update

CMPSO uses an external archive to store the non-dominated solutions from all the

swarms. This archive is not only used to store the non-dominated solutions to be reported at

the end of the evolution like in traditional MOP algorithm [209], but also is used for

information sharing among different swarms. Therefore, the particles in all the swarms can

access the information in the archive and use it to guide the flying, as indicated in Eq. (4-4)

Denoted as A, the archive is initialized to be empty and is updated in every generation.

Researches show that it is better to use an archive with a fixed maximal size because the


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number of non-dominated solutions may increase very fast. Therefore, in CMPSO, the

archive is set to with maximal size of NA and the size of the archive in the current generation

is denoted as na. In the end of every generation, the archive A is updated. The archive update

process is described as follow:

Step 1: a new set S is initialized to be empty. Then the pBest of each particle in each

swarm is added into the set S.

Step 2: All the solutions in the old archive A are added into the set S.

Step 3: The ELS is performed on each solution in the archive A to form a new solution.

All the new solutions are then added into the set S. After the above operations, the set S

would have (N×M+2×na) solutions, where N and M are the population size of a swarm

and the swarm number (objectives number) respectively, and na is the number of

solutions of the old archive A.

Step 4: A non-dominated solutions determining procedure is performed on the set S to

determine all the non-dominated solutions and store them in a set R.

Step 5: If the size of R (the number of the non-dominated solutions) is not larger than NA,

then all the non-dominated solutions are stored in the new archive A, and na is set as the

size of R. Otherwise, all the non-dominated solutions are sorted according to the density,

and the first NA less crowded ones are selected to store in the new archive A, with the

number na set as NA.

The following parts give the details of the elitist learning strategy procedure, the

non-dominated solutions determining procedure, and the density based selection procedure.

1） Elitist Learning Strategy

The ELS was first introduced in the adaptive PSO in Chapter 2.3.3 for the globally best

particle to jump out of possible local optima. In this chapter, we perform the ELS on all the

solutions in the archive because they are all globally best solutions of CMPSO. The

pseudo-code of the procedure is presented in Fig. 4-2(a) and is described as follows

For each solution Ai in the archive A, let the new solution Ei first equals to Ai, and then a

random dimension d is selected to perform the Gaussian perturbation as

Aid = Aid + (Xdmax –Xdmin)Gaussian(0, 1) (4-6)

where Xmax,d and Xmin,d are the upper and lower bounds of the dth dimension respectively.

Gaussian(0, 1) is a random value generated by a Gaussian distribution with mean value 0 and

standard deviation 1.


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After the perturbation, the Eid is checked and is guaranteed to be in the search range

[Xmin,d, Xmax,d], otherwise, Eid is set to the corresponding bound. Later, all the objectives of the

new solution Ei are calculated, and the solution Ei is added into the set S.

Fig. 4-2 The archive update process.

2） Non-dominated Solutions Determining

This procedure is to determine the non-dominated solutions in a given solutions set S.

The procedure is described as follows and its pseudo-code is given in Fig. 4-2(b). A set R is

used to store the non-dominated solutions and it is initialized to be empty. Then for each

solution i in the set S, the procedure checks whether the solution i is dominated by any other

solution j. If solution i is not dominated by any other solution, then solution i is added into

the set R. This checking process is performed on all the solutions in the set S and all the

non-dominated solutions can be determined and stored in the set R.

3） Density Based Selection


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This procedure is performed if the size of R is larger than the maximal size of the

archive, NA. The function is to select less crowded solutions in the first NA into the new

archive. For the detailed calculation, the procedure is given in Fig. 4-2(c) and is described as

follows. Given a set of solutions R, the distance of each solution is initialized to be zero. Then

all the solutions are sorted according to each objective value, from the smallest to the largest.

For each objective, the boundary solutions, that is, the solutions with the smallest value and

the largest value on this objective are assigned an infinite distance value. The distance of the

other solutions is increased by the absolute normalized difference of the objective values

between the two adjacent solutions.

After the density estimation, all the solutions in the set R are assigned with a distance.

Then we can select the first NA solutions with large distances to the new archive A.

4.3.3 Complete CMPSO

With the MPMO framework, CMPSO adopts PSO for each swarm and achives M

objectives by using M co-evolutionary population to solve MOP. The complete CMPSO

algorithm is given in Fig. 4-3.

Fig. 4-3 The complete flowchart of CMPSO.


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4.3.4 Complexity Analysis of CMPSO

The complexity analysis of CMPSO is described as follow:

Firstly, according to the flowchart illustrated in Fig. 4-3, the calculation process of

CMPSO is smilar to the traditional PSO except that CMPSO uses M population. Assume that

the population size of each population is N, then the compuatation complexity of all particle

in each generation is O1=O(N×M).

Secondly, with the help of Fig. 4-2, we can see that the computation complexity of

archive update operation in each generation mainly includes 3 parts as follow:

A) Assume that the size of the set S is |S|, then the computation complexity of the

construction of S is Oa(|S|).

B) Another important step in archive update operation is to determine the

non-dominated solutions in a given solutions set S and store them in a set R as shown in Fig.

4-2(b). The computation complexity of this step is Ob(|S|2).

C) The final step in archive update is the density based selection operation shown in

Fig. 4-2(c). As the operation is performed on R, assume that the size of R is |R|, then the

complexity of this step is Oc=O(M×|R|2)+O(|R|2), where O(M×|R|2) and O(|R|2) represent the

complexity of the distance calculation and the sort operation based on distance, respectively.

Therfore, the complexity of archive update is

O2=Oa(|S|)+Ob(|S|2)+Oc(M×|R|2)=O(|S|2)+O(M×|R|2). On the other hand, since both of |S| and

|R| are linear with the size of archive NA, O2 can be further represented as O2=O(M×NA2).

Overall, the complexity of CMPSO in each generation can be calculated as

OCMPSO=O1+O2=O(N×M)+O(M×NA2)=O(M×(N+NA2)). Due to the comparability of the

population size and archive size, OCMPSO can be transformed to

OCMPSO=O(M×(N+NA2))=O(M×NA2). If the maximum generation of CMPSO is G, then the

complexity of CMPSO is OCMPSO=G×O(M×NA2)=O(G×M×NA2).

From the above analysis, we can see that the complexity of CMPSO is mainly focused

on the density based selection process, that is O(M×NA2). In other multi-objective algorithms

such as NSGA-II, the algorithm complexity mainly lies on the non-dominated solutions

sorting process. Hence, the complexity of NSGA-II is ONSGA-II=G×O(M×N2) with M

objectives and population size N [184].

From the complexity expressions of CMPSO and NSGA-II, we see that the complexity

of the two algorithms can both be expressed as O(G×M×NA2) or O(G×M×N2) due to the the

comparability of the population size N and archive size NA. That is to say, although using


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multiple population, the complexity of CMPSO is still not higher than the other traditional

multi-objective evolutionary algorithms.

4.4 Experimental Veerification and Comparisons

4.4.1 Test Problems

Various test problems have been proposed to evaluate the multi-objective optimization

algorithms in the literature. First, we adopt the most frequently used problems ZDT1, ZDT2,

ZDT3, ZDT4, and ZDT6 from the ZDT test set [210]. However, some researchers argue that

ZDT problems are lack of characteristics such as variables linkage, objective functions

modality, and PF shape. Therefore, we also adopt DTLZ1 and DTLZ2 from the DTLZ test set

[211], and WFG1, WFG2, WFG3, and WFG4 from the WFG test set [212], which are with

multimodal objective functions and non-separable variables. Most recently, Zhang et al. [213]

proposed a new UF test set where the problems are with complicated Pareto sets. In this

chapter, we further choose all the two-objective unconstrained problems UF1-UF7 to

evaluate the algorithm performance Table 4-1 Characteristics of the Test Problems

Name Dimension Search Range Global Optima Note ZDT1 30 xi∈[0, 1], 1≤i≤D x1∈[0, 1], xi=0, 2≤i≤D Convex, F1 U, F2 U (Unimodal) ZDT2 30 xi∈[0, 1], 1≤i≤D x1∈[0, 1], xi=0, 2≤i≤D Concave, F1 U, F2 U ZDT3 30 xi∈[0, 1], 1≤i≤D x1∈[0, 1], xi=0, 2≤i≤D Convex, Disconnected, F1 U, F2 M (Multimodal)

ZDT4 10 x1∈[0, 1], xi∈[-5,5], 2≤i≤D x1∈[0, 1], xi=0, 2≤i≤D Concave, F1 U, F2 M

ZDT6 10 xi∈[0, 1], 1≤i≤D x1∈[0, 1], xi=0, 2≤i≤D Concave, Non-uniformly, F1 M, F2 M DTLZ1 10 xi∈[0, 1], 1≤i≤D x1∈[0, 1], xi=0.5, 2≤i≤D Linear, F1 M, F2 M DTLZ2 10 xi∈[0, 1], 1≤i≤D x1∈[0, 1], xi=0.5, 2≤i≤D Concave, F1 U, F2 U WFG1 10 zi∈[0, 2i], 1≤i≤D zi∈[0, 2i], 1≤i≤k, zi=0.35, k+1≤i≤D Convex, Mixed, F1 U, F2 U WFG2 10 zi∈[0, 2i], 1≤i≤D zi∈[0, 2i], 1≤i≤k, zi=0.35, k+1≤i≤D Convex, Disconnected, F1 U, F2 M WFG3 10 zi∈[0, 2i], 1≤i≤D zi∈[0, 2i], 1≤i≤k, zi=0.35, k+1≤i≤D Linear, Degenerated, F1 U, F2 U WFG4 10 zi∈[0, 2i], 1≤i≤D zi∈[0, 2i], 1≤i≤k, zi=0.35, k+1≤i≤D Concave, F1 M, F2 M

UF1 30 x1∈[0, 1], xi∈[-1,1], 2≤i≤D

x1∈[0, 1], 1sin(6 / )ix x j Dπ π= + , 2≤i≤D Convex, F1 M, F2 M

UF2 30 x1∈[0, 1], xi∈[-1,1], 2≤i≤D

x1∈[0, 1], 21 1 1 1 121 1 1 1 2

(0.3 cos(24 4 / ) 0.6 )cos(6 / ),

(0.3 cos(24 4 / ) 0.6 )sin(6 / ),i

x x i D x x j D j Jx

x x i D x x j D j J

π π π ππ π π π

⎧ + + + ∈⎪= ⎨+ + + ∈⎪⎩

,

2≤i≤D

Convex, F1 M, F2 M

UF3 30 xi∈[0,1], 1≤i≤D x1∈[0, 1], 0.5(1.0 (3( 2)) /( 2))1

i Dix x + − −= , 2≤i≤D Convex, F1 M, F2 M

UF4 30 x1∈[0, 1], xi∈[-2,2], 2≤i≤D

x1∈[0, 1], 1sin(6 / )ix x j Dπ π= + , 2≤i≤D Concave, F1 M, F2 M

UF5 30 x1∈[0, 1], xi∈[-1,1], 2≤i≤D

(F1, F2) = ( / 2i N , 1– / 2i N ), 0≤i≤2N, N=10 Scatter, F1 M, F2 M

UF6 30 x1∈[0, 1], xi∈[-1,1], 2≤i≤D 1 1

[(2 1) / 2 ,2 / 2 ]N

iF i N i N

== ∪ − , F2=1–F1, N=2 Linear, Disconnected, F1 M, F2 M

UF7 30 x1∈[0, 1], xi∈[-1,1], 2≤i≤D

x1∈[0, 1], 1sin(6 / )ix x j Dπ π= + , 2≤i≤D Linear, F1 M, F2 M


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Totally, 18 test problems (5 ZDT, 2 DTLZ, 4 WFG, and 7 UF) are used and their

characteristics are described in Table I. The problems are with characteristics of different

dimensionalities like 10 and 30, of different objective functions like unimodal and

multimodal, of different PFs like scatter, linear, convex, concave, disconnected,

non-uniformly, and mixed, and of different Pareto set shapes like simple and complicated.

Therefore, they are comprehensive and useful for testing the algorithm’s performance from

different aspects. For more details of the problems, please refer to [210], [211], [212], and

[213] for ZDT, DTLZ, WFG, and UF respectively.

4.4.2 Performance Metric

The inverted generational distance (IGD) indicator is adopted in this chapter as the

performance metric because it can reflect both the convergence and diversity of the obtained

solutions to the true PF. The indicator has been widely adopted and strongly recommended in

MOP community in recent years [213]. Assume that the set of non-dominated solutions

obtained by an algorithm is A and a set of solutions uniformly sampled along the true PF is P,

the calculation of IGD(A, P) is as

PAPd

PAIGDP

i i∑ ==||

1),(

),( (4-7)

where |P| is the size of the set P and d(Pi, A) denotes the distance between the solution Pi and

the solution in the set A that is nearest to Pi, measured by the Euclidean distance in the

objective space. This IGD indicator has an assumption that the true PF is known. In this

chapter, we sample 500 uniformly distributed points along the PF to form the set P for each

problem. Intuitively, if the non-dominated solutions in the set A have a good spread along the

true PF, then the indicator IGD will have a small value.

4.4.3 Experimental Settings

In this chapter, we compare the results obtained by CMPSO with not only MOEAs, but

also MOPSOs. The MOEAs include NSGA-II [184], generalized DE 3 (GDE3) [206], and

MOEA with decomposition and DE operators (MOEA/D-DE) [207], while the MOPSOs

include multi-objective comprehensive learning PSO (MOCLPSO) [214], optimized MOPSO

(OMOPSO) [215], and VEPSO [216]. These algorithms are chosen because NSGA-II and

MOCLPSO are two state-of-the-art algorithms, GDE3 and MOEA/D-DE are two most


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recently well performing MOEAs, OMOPSO is a very salient MOPSO according to a very

recent comparative study [217], and VEPSO is a MOPSO that also uses multiple populations.

Therefore, these algorithms are representative and helpful to make the comparisons more

comprehensive and convincing

The parameters of the above algorithms are set according to the proposals in their

corresponding references, as summarized in Table 4-2. For CMPSO, we adopt the common

configurations that the inertia weight in Eq. (4-4) linearly decreasing from 0.9 to 0.4, and the

acceleration coefficients c1, c2, and c3 setting to be 4.0/3. We set such a value for ci because

that the sum of ci usually is to be 4.0 in PSO. As the CMPSO uses different swarms to

optimize different objectives, we set a relative small population size of 20 for each swarm. In

order to make the comparisons fair, all the seven algorithms have the same archive size of

100. The non-dominated solutions in the archive are updated and used to calculate the IGD

value in every generation according to Eq. (4-7) and are reported at the end of the algorithm

running.

It should be noticed that when solving different kinds of MOPs, different population size

and different maximal number of function evaluations (FEs) are used [184][213]. The

population sizes in Table 4-2 are used when solving the ZDT problems and the maximal

number of FEs is set to be 25000 [184]. However, when solving the more difficult DTLZ and

WFG problems, the population size is set to be 200 for all the algorithms (except CMPSO

which still uses population size of 20 for each swarm) and the maximal number of FEs is

1×105. When solving the complicated UF problems, all the algorithms (CMPSO still uses

population size of 20 for each swarm) are with population size of 300 and the maximal

number of FEs of 3×105 [213]. The impacts of population size on the CMPSO performance

will be investigated in Section 4.4.8. Moreover, the experimental results are the average

values of 30 independent runs. The best results are denoted by the bold font. The results

obtained by different algorithms are compared with the CMPSO by Wilcoxon’s rank sum test

with significant level α=0.05.

Table 4-2 Parameters Settings of the Algorithms

Algorithm Parameters Settings NSGA-II N=100, px=0.9, pm=1/D, ηc=20, and ηm=20

GDE3 N=100, CR=0.0, and F=0.5 MOEA/D-DE N=100, CR=1.0, F=0.5, η=20, pm==1/D, T=20, δ=0.9, and nr=2 MOCLPSO N=50, pc=0.1, pm=0.4, ω=0.9→0.2, and c=2.0 OMOPSO N=100, ω=rand(0.1, 0.5), c1=rand(1.5, 2.0), c2=rand(1.5, 2.0)

VEPSO N=100, χ=0.729, c1=c2=2.05, and M=6 CMPSO N=20, ω=0.9→0.4, and c1=c2=c3=4.0/3


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4.4.4 Experimental Results on ZDT Problems

Table 4-3 compares the results on the ZDT problems. The results show that CMPSO is

promising in dealing with ZDT problems with both convex and concave PFs. It performs best

on ZDT1 which is with convex PF and on ZDT2 which is with concave PF. It also performs

the second best on ZDT6 (only slightly worse than MOCLPSO) which is with non-uniform

concave PF. Moreover, CMPSO is very promising (the third best) on ZDT3 whose PF is

disconnected convex. Table 4-3 Results Comparisons on the ZDT Problems

Function NSGA-II GDE3 MOEA/D-DE MOCLPSO OMOPSO VEPSO CMPSO

ZDT1 Mean 5.00×10-3 1.27×10-2 0.16 4.80×10-3 7.02×10-3 0.31 4.13×10-3

Std 2.33×10-4 1.56×10-3 1.93×10-2 1.76×10-4 7.83×10-4 3.39×10-2 8.30×10-5

Rank 3 − 5 − 6 − 2 − 4 − 7 − 1

ZDT2 Mean 0.19 2.97×10-2 0.23 0.38 6.06×10-3 0.33 4.32×10-3

Std 0.28 1.82×10-2 3.07×10-2 0.30 3.81×10-4 0.11 1.03×10-4

Rank 4 − 3 − 5 − 7 − 2 − 6 − 1

ZDT3 Mean 1.54×10-2 1.16×10-2 0.23 5.49×10-3 2.30×10-2 0.60 1.39×10-2

Std 2.71×10-2 2.24×10-3 2.17×10-2 2.49×10-4 5.99×10-3 9.40×10-2 3.49×10-3

Rank 4 − 2 + 6 − 1 + 5 − 7 − 3

ZDT4 Mean 0.29 0.34 0.31 3.26 16.47 26.75 0.79 Std 0.40 0.37 0.23 1.35 4.12 6.06 0.26

Rank 1 + 3 + 2 + 5 − 6 − 7 − 4

ZDT6 Mean 6.22×10-3 7.36×10-2 1.54 3.69×10-3 4.61×10-3 0.41 3.72×10-3

Std 7.02×10-4 9.16×10-2 0.13 1.31×10-4 3.36×10-4 0.19 1.47×10-4

Rank 4 − 5 − 7 − 1 ≈ 3 − 6 − 2 Final Rank

Total 16 18 26 16 20 33 11 Final 2 4 6 2 5 7 1

Better−Worse -3 -1 -3 -2 -5 -5 ‘+’、‘−’ and ‘≈’ indicate that the results of the algorithm are significant better than, worse than, and similar to ones of CMPSO by Wilcoxon’s

rank sum test with α=0.05. As CMPSO has the best performance on ZDT1 and ZDT2 whose objectives are all

unimodal, it indicates that CMPSO has the strong convergence ability to approximate the PF

of MOP with simple objectives. Table 4-3 also shows that all the MOPSOs are beaten by

MOEAs on ZDT4. This may be caused by the local PFs of ZDT4 for that it is with the

multimodal Rastrigin function. However, CMPSO is still the best algorithm among all the

four MOPSOs and only CMPSO can obtain comparable results with MOEAs. This indicates

that CMPSO has the ability to avoid local PFs caused by complex objectives. Moreover,

when the general performance is considered over all the problems, CMPSO is the winner for

that it has the first average rank among the seven contenders over all the five problems. The

Wilcoxon’s rank sum tests also indicate that CMPSO significantly outperforms all the six

competitors on the ZDT set problems.

Fig. 4-4 visualizes the final non-dominated solutions obtained by different algorithms in

all the 30 runs when solving some of the ZDT problems. Notice that some algorithms have


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similar performance on the same problems, and therefore we only use one algorithm as the

representative. For example, the figures of the solutions obtained by NSGA-II, GDE3,

MOCLPSO, OMOPSO, and CMPSO on ZDT1 are not evidently different, and therefore we

only use the solutions obtained by CMPSO to plot in Fig. 4-4(a) for clarity. The figures in Fig.

4 show that the solutions obtained by CMPSO not only approximate the whole PF well, but

also form a good spread along the whole PF. Fig. 4-4(c) and Fig. 4-4(d) show the obtained

solutions to ZDT4. As MOCLPSO, OMOPSO, and VEPSO perform very poorly on this

problem, the solutions are not plotted in the figures.

Fig. 4-4 The final non-dominated solutions of the ZDT problems in all the 30 runs.

4.4.5 Experimental Results on DTLZ and WFG Problems

The results on the DTLZ and WFG problems are presented and compared in Table 4-4.

The results show that CMPSO performs competitively with GDE3, NSGA-II, and

MOEA/D-DE on DTLZ1 and performs the best on DTLZ2. Moreover, CMPSO yield the best

IGD values on all the four WFG problems. In general, CMPSO has the first average rank

over all the six DTLZ and WFG problems, followed by other MOPSOs and then MOEAs.

The statistics by the Wilcoxon’s rank sum tests also confirm that CMPSO has significant

better performance than all the six contenders. As these problems are with mixed,

disconnected, or degenerated PFs, the good performance of CMPSO indicates that it is

promising not only in MOPs with simple PFs like the ZDT problems, but also in MOPs with

complex PFs like the DTLZ and WFG problems.


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Fig. 4-5 further confirms the advantages of CMPSO by plotting the obtained

non-dominated solutions in the 30 runs. Although all the algorithms fail to obtain the true PF

of WFG1, CMPSO gives the best approximate to PF and yields the best diversity to spread

along the PF. For WFG2, WFG3, and WFG4, only CMPSO obtain solutions approximating

to the PFs. CMPSO is observed to perform remarkably better than not only the other

MOPSOs, but also all the compared MOEAs. Table 4-4 Results Comparisons on the DTLZ and WFG Problems


DTLZ1 Mean 2.75×10-3 2.42×10-3 2.75×10-3 38.75 43.04 91.56 5.67×10-2

Std 2.86×10-4 1.34×10-4 4.36×10-4 7.36 8.06 23.42 2.21×10-2

Rank 2 + 1 + 3 + 5 − 6 − 7 − 4

DTLZ2 Mean 5.81×10-3 5.49×10-3 3.33×10-2 8.79×10-3 6.65×10-3 6.07×10-2 4.62×10-3

Std 4.70×10-4 6.85×10-4 3.02×10-3 8.06×10-4 6.08×10-4 8.69×10-3 1.50×10-4

Rank 3 − 2 − 6 − 5 − 4 − 7 − 1

WFG1 Mean 1.80 2.58 2.57 1.37 1.38 1.64 1.23 Std 0.31 1.03×10-2 1.62×10-3 0.13 3.71×10-3 0.30 6.69×10-2

Rank 5 − 7 − 6 − 2 − 3 − 4 − 1

WFG2 Mean 0.46 1.14 0.96 0.38 0.40 0.41 0.11 Std 4.03×10-2 4.66×10-2 7.94×10-2 3.45××10-2 4.65×10-2 4.34×10-2 6.19×10-2

Rank 5 − 7 − 6 − 2 − 3 − 4 − 1

WFG3 Mean 0.36 1.66 0.78 0.30 0.30 0.33 1.47×10-2

Std 3.07×10-2 0.28 4.83×10-2 2.36×10-2 2.61×10-2 2.25×10-2 5.80×10-4

Rank 5 − 7 − 6 − 2 − 3 − 4 − 1

WFG4 Mean 0.35 1.09 0.65 0.22 0.22 0.27 1.37×10-2

Std 4.17×10-2 0.12 3.34×10-2 1.00×10-2 1.23×10-2 1.68×10-2 4.99×10-4

Rank 5 − 7 − 6 − 2 − 3 − 4 − 1 Final Rank

Total 25 31 33 18 22 30 9 Final 4 6 7 2 3 5 1

Better − Worse -4 -4 -4 -6 -6 -6

Fig. 4-5 The final non-dominated solutions of the WFG problems in all the 30 runs.


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4.4.6 Experimental Results on UF Problems

The above two sub-sections have demonstrated that CMPSO shows a very good

performance on ZDT problems and the advantages of CMPSO become more evident on

DTLZ and WFG problems as the objective functions and the PFs become more complex. In

this sub-section, the algorithms performance is further compared on the UF problems, which

are the recently proposed problems with complicated Pareto set.

The results compared in Table 4-5 show that CMPSO performs best on UF2, UF4, and

UF5 while MOEA/D-DE performs best on UF3 and UF7, NSGA-II on UF6, and GDE3 on

UF1. Even though GDE3 and MOEA/D-DE outperform CMPSO on UF1, the results are not

significantly different according to the Wilcoxon test. In general, CMPSO has the first

average rank, followed by GDE3 and MOEA/D-DE over all the 7 UF problems. By the

Wilcoxon’s rank sum tests, CMPSO is also the most promising algorithm in solving the UF

set problems. Table 4-5 Results Comparisons on the UF Problems


UF1 Mean 7.30×10-2 5.75×10-2 5.96×10-2 0.10 9.81×10-2 0.71 6.64×10-2

Std 2.46×10-2 2.48×10-2 2.15×10-2 7.17×10-3 7.91×10-3 0.15 1.99×10-2

Rank 4 ≈ 1 ≈ 2 ≈ 6 − 5 − 7 − 3

UF2 Mean 2.06×10-2 2.02×10-2 6.63×10-2 0.11 7.24×10-2 0.15 1.69×10-2

Std 3.67×10-3 3.81×10-3 1.32×10-2 3.39×10-3 3.54×10-3 1.29×10-2 3.37×10-3

Rank 3 − 2 − 4 − 6 − 5 − 7 − 1

UF3 Mean 6.95×10-2 0.16 3.89×10-2 0.48 0.37 0.58 9.80×10-2

Std 1.14×10-2 6.66×10-2 1.57×10-2 1.55×10-2 9.71×10-3 4.83×10-2 1.39×10-2

Rank 2 + 4 − 1 + 6 − 5 − 7 − 3

UF4 Mean 4.26×10-2 2.95×10-2 4.72×10-2 0.12 0.16 0.17 2.38×10-2

Std 4.46×10-4 1.03×10-3 1.59×10-3 1.10×10-2 1.39×10-2 6.22×10-3 1.90×10-3

Rank 3 − 2 − 4 − 5 − 6 − 7 − 1

UF5 Mean 0.32 0.21 0.33 0.51 0.74 3.25 0.20 Std 8.41×10-2 1.61×10-2 5.41×10-2 0.18 0.12 0.53 2.01×10-2

Rank 3 − 2 − 4 − 5 − 6 − 7 − 1

UF6 Mean 0.12 0.30 0.14 0.40 0.40 2.83 0.14 Std 1.93×10-2 1.72×10-2 9.05×10-2 4.30×10-2 3.40×10-2 0.78 2.04×10-2

Rank 1 + 4 − 3 − 6 − 5 − 7 − 2

UF7 Mean 0.16 2.97×10-2 8.34×10-3 0.19 0.22 0.69 0.12 Std 0.16 1.02×10-3 9.40×10-4 0.15 0.15 0.16 0.13

Rank 4 ≈ 2 + 1 + 5 − 6 − 7 − 3 Final Rank

Total 20 17 19 39 38 49 14 Final 4 2 3 6 5 7 1

Better − Worse -1 -4 -2 -7 -7 -7 Algorithms NSGA-II GDE3 MOEA/D-DE MOCLPSO OMOPSO VEPSO CMPSO


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Fig. 4-6 The final non-dominated solutions of the UF problems in all the 30 runs.


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Fig. 4-6 further compares MOEA/D-DE with CMPSO by plotting the 30 non-dominated

fronts obtained by the two algorithms. MOEA/D-DE is observed to be weaker than our

CMPSO on UF1 and UF2 for that MOEA/D-DE misses the solutions located in the middle of

the PFs. For UF3, CMPSO has difficulty in obtaining enough good solutions along the PF as

that by MOEA/D-DE. However, CMPSO seems to be more stable than MOEA/D-DE on UF4

because the solutions obtained by CMPSO locate closer to the true PF of UF4, as shown in

Fig. 4-6(g) and Fig. 4-6(h)

4.4.7 The Benefit of Shared Archive

The CMPSO algorithm uses an external shared archive to let different swarms share

their search information and communicate with each other efficiently. In this sub-section, the

benefit of shared archive is investigated, including the benefit of the ELS used in the archive

update process, and the benefit of using the shared archive solutions information in the

particle update equation. The experimental results are given in Table 4-6. Table 4-6 Comparisons Between CMPSO and Its Variants CMPSO-non-ELS (CMPSO without ELS in the

Archive Update) and CMPSO-non-aBest (CMPSO without Using Archive Information for Particle Update)

Function CMPSO CMPSO-non-ELS CMPSO-non-aBest ZDT1 4.13×10-3 0.30 1.09×10-2 ZDT2 4.32×10-3 0.84 1.81×10-2 ZDT3 1.39×10-2 0.47 2.42×10-2 ZDT4 0.79 26.09 0.78 ZDT6 3.72×10-3 0.18 6.63×10-2

DTLZ1 5.67×10-2 1.43×102 6.36×10-2 DTLZ2 4.62×10-3 0.12 4.64×10-3 WFG1 1.23 2.19 1.66 WFG2 0.11 0.57 0.10 WFG3 1.47×10-2 0.42 1.48×10-2 WFG4 1.37×10-2 0.31 1.37×10-2 UF1 6.64×10-2 0.38 5.36×10-2 UF2 1.69×10-2 0.16 1.61×10-2 UF3 8.90×10-2 0.57 7.76×10-2 UF4 2.38×10-2 0.19 2.35×10-2 UF5 0.20 2.53 0.19 UF6 0.14 1.20 0.16 UF7 0.12 0.46 0.12

As the ELS is demonstrated to be helpful for bringing in diversity to avoid being trapped

into local optima when solving single objective optimization problems, it is also expected that

the ELS can help CMPSO to avoid local PFs in MOP. The comparisons in Table 4-6 show

that CMPSO significantly outperforms its variant without using the ELS in the archive update


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(denoted as CMPSO-non-ELS) on all the 18 test problems. The advantages of CMPSO are

more evident when solving MOPs with multimodal objective functions for that

CMPSO-non-ELS is easy to be trapped into local PFs. We further compare the results

obtained by CMPSO and CMPSO-non-ELS on the WFG2 (with disconnected PF and with

one multimodal objective function), WFG4 (with concave PF and with two multimodal

objective functions), and UF2 (with convex PF and with two multimodal objective functions)

in Fig. 4-7. These figures clearly show that CMPSO can approximate the true PFs of these

problems whilst CMPSO-non-ELS is totally trapped into local PFs

Fig. 4-7 The final non-dominated solutions found by CMPSO and CMPSO-non-ELS in all the 30 runs.

The benefit of using the shared archive information to guide the particle update is also

summarized in Table 4-6. The shared archive information is expected to be beneficial for

accelerating the convergence speed to approximate the PF. For the MOP with unimodal PF,

CMPSO is observed to remarkably outperform its variant without using archived information

in the particle update (denoted as CMPSO-non-aBest), e.g., on most of the ZDT, DTLZ, and

WFG problems. However, CMPSO-non-aBest seems to be better than CMPSO on some of

the UF problems. The reason may be that these UF problems have complicated Pareto sets

and therefore algorithms with too fast convergence speed will cause premature convergence

and cannot search the whole space more efficiently.

Fig. 4-8 compares the convergence characteristics of the IGD indicator on ZDT1 (with

uniform convex PF and unimodal objective functions), ZDT6 (with non-uniform concave PF

and multimodal objective functions), WFG1 (with mixed convex PF and unimodal objective

functions), and UF2 (with convex PF and multimodal objective functions) during the

CMPSO and CMPSO-non-aBest search processes. The figures further show that the

utilization of the archived information to guide the particle update remarkably accelerates the

convergence speed for the algorithm to approximate the PF, especially on problems with

unimodal objective functions. However, when the objective functions are multimodal, such as

Fig. 4-8(d) for UF2, CMPSO is faster in the early evolutionary phase, but is taken over by


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CMPSO-non-aBest in the late evolutionary phase. This is because that too fast convergence

speed may cause an adverse effect on prematurity and make the algorithm not search the

whole space sufficiently when the Pareto sets are complicated. Therefore, in the late

evolutionary phase, CMPSO may perform slightly poorly than CMPSO-non-aBest does on

some UF problems.

Fig. 4-8 The mean IGD of CMPSO and CMPSO-non-aBest during the evolutionary process.

In order to show more clearly how CMPSO can approximate the PF faster than

CMPSO-non-aBest does, we plot the finial non-dominated solutions of them in Fig. 4-9 for

UF1, UF2, and UF3. These solutions are obtained after 1000 FEs, and are obtained by the run

that has the minimal IGD value among the 30 runs. The figures confirm that CMPSO

approximates the PF faster than CMPSO-non-aBest in the early evolutionary phase.

Fig. 4-9 The final non-dominated solutions found by CMPSO and CMPSO-non-aBest after 1000 FEs.


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4.4.8 Impacts of Parameter Settings

1) Population Size for Each Swarm

As each swarm optimizes only one objective in CMPSO, we set a relative small

population size of 20 particles for each swarm. Herein, the population size for each swarm is

set to be 40, 60, 80, and 100, respectively, to investigate the impact of population size on

algorithm’s performance. The investigations are conducted on ZDT1 whose objective

functions are umimodal and on UF1 whose objective functions are multimodal. For CMPSO

with different population size, the other parameters are still the same as in Chapter 4.4.3 and

the maximal number of FEs is still 25000 for ZDT1 and 3×105 for UF1.

The average values of 30 independent runs on the IGD indicator are compared in Fig.

4-10. The comparisons show that small population size is efficient enough for CMPSO to

obtain good performance, especially for simple MOP. This may be due to the contribution of

the MPMO technique in reducing the search complexity for each swarm because only one

objective is optimized by each swarm. When the population size increases to be large,

CMPSO performs even worse, e.g., on the ZDT1 problem. This may be caused that with the

fixed value of maximal number of FEs, larger population size reduces the evolutionary

generation, and at last affects the algorithm’s performance. However, when solving

complicated MOP, increasing the population size can increase the diversity of the algorithm

and therefore can lead to better results, e.g. on the UF1 problem. Nevertheless, too large

population size costs much computational burden in each generation and therefore may

weaken the performance when the maximal number of FEs is fixed. By considering both the

computational burden and the performance, this chapter adopts the population size of 20 for

each swarm. In general, population size of 20-60 for each swarm may be promising. The

small population size can bring good performance is an advantage of the CMPSO algorithm.

20 40 60 80 1000.0041

0.0042

0.0043

0.0044

0.090

0.095

0.100

0.105

0.110

0.115

Mea

n IG

D in

dica

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alue

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ZDT1 UF1

Fig. 4-10 The mean IGD of CMPSO with different population size.


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2) Maximal Number of FEs

As shown in Section 4.4.4, CMPSO and other MOPSOs perform poorly on ZDT4. It

could be due to that ZDT4 is with multimodal objective functions while CMPSO does not

have sufficient number of FEs to converge to the true PF. Herein, we keep the other

parameters the same as in Section 4.4.3 and set different maximal number of FEs (e.g., 1×105,

2×105, and 3×105) for CMPSO to solve ZDT4. The final non-dominated solutions found with

different maximal number of FEs are plotted in Fig. 4-11.

When compared with Fig. 4-4(c), it is clear that by increasing the maximal number of

FEs, CMPSO can search the space sufficiently to approximate the true PF well. Moreover,

we plot the solutions obtained by OMOPSO in all the 30 runs in Fig. 4-11. When compared

with CMPSO, it is clear that CMPSO has stronger global search ability than OMOPSO to

approximate the PF. This indicates that the MPMO technique is helpful to enhancing the

MOPSOs’ performance.

Fig. 4-11 The final non-dominated solutions found by CMPSO and OMOPSO with different FEs on ZDT4.

3) Inertia Weight ω and Acceleration Coefficients ci

To investigate the impact of ω on the MOPSOs’ performance, we test different values of

ω (e.g., 0.1, 0.3, 0.5, 0.7, and 0.9) on CMPSO, MOCLPSO, and OMOPSO. The

investigations are conducted on DTLZ2 with umimodal objective functions and on UF1

multimodal objective functions.

Fig. 4-12(a) and (b) show the mean IGD value of DTLZ1 and UF1 when MOPSOs using

different inertia weight values. For DTLZ2 whose objective functions are unimodal, a relative

small ω value would be preferred while for UF1 whose objective functions are multimodal, a

relative large ω value seems to be preferred. This may be due to that large ω is helpful for

global search while small ω is beneficiary for local fine tuning. The results are also compared

with the ones obtained by the MOPSOs using ω values as in Table 4-2. The figure shows that

the parameters in Table II are promising. Moreover, it is evident from Fig. 4-12(a) and (b)


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that CMPSO is less sensitive to the ω value while the other two MOPSOs, especially

MOCLPSO, are affected by the ω value significantly. This is another advantage of the

CMPSO algorithm.

The impact of the acceleration coefficients ci on the CMPSO performance is also

investigated on DTLZ2 and UF1, with the results shown in Fig. 4-12(c) and (d). The results

further confirm that the parameters in Table 4-2 are promising and CMPSO is much less

sensitive to the ci value when compared with MOCLPSO and OMOPSO, showing the

advantage of CMPSO.

It follows that the performance of CMPSO is not dependent on the parameters, and the

parameter values used in this chapter are also widely adopted in PSO. Therefore, CMPSO

retains the simplicity and easy implement of PSO, as well as improves the performance of

PSO on multi-objective optimization problems by coevolutionary technique with a multiple

population for multiple objectives framework, which is inspired by the idea of ensemble

learning in machine learning.

0.00

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MOCLPSO OMOPSO CMPSO

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0.9--->0.40.90.70.50.3

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0.1

(a) Different ω on DTLZ2 (b) Different ω on UF1

0.000

0.005

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4.0/3

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3.53.02.52.01.51.00.5Acceleration coefficient value

(c) Different ci on DTLZ2 (d) Different ci on UF1

Fig. 4-12 The mean IGD on DTLZ2 and UF1 of MOPSOs with different ω and different ci.


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4.5 Chapter Summary

Learning from the ensemble learning in machine learning, this chapter proposed a novel

technique termed multiple populations for multiple objectives.

The advantages and characteristics of the proposed MPMO and CMPSO algorithm are

as follow:

1) As each swarm focus on optimizing one objective, it can use the conventional or any

other improved PSO to solve a single objective problem. Importantly, the difficulty of

fitness assignment can be avoided.

2) As an external shared archive is used to store the non-dominated solutions found by

different swarms and the shared archive information is used to guide the particles update,

the algorithm can use the whole search information to approximate the whole PF fast.

3) As an ELS is performed on the archived solutions in the update process, the algorithm is

able to avoid local PFs. This is helpful for MOPs with multimodal objective functions or

with complicated Pareto sets.

4) As the experiments demonstrated that the parameters have less significant impact on the

performance of the CMPSO compared with other MOPSO algorithms, CMPSO is still

simple and easy to use. Hence, the CMPSO has important application value and promoted

meaning.

In a conclusion, inspired by the ensemble learning in machine learning, this chapter

designed a new co-evolutionary technique named MPMO, and proposed a co-evolutionary

multi-swarm PSO with shared archive based on the MPMO technique. Experimental results

show the effectiveness and efficiency of the MPMO framework and CMPSO algorithm. The

design of MPMO framework and CMPSO algorithm for multiobjective problems is an

important and successful exploration of the machine learning aided PSO design.

Chapter 5 Orthogonal Learning Particle Swarm Optimization for Power Electronic Circuit Optimization with Free Search Range

103

Chapter 5 Orthogonal Learning Particle Swarm Optimization for

Power Electronic Circuit Optimization with Free Search Range

5.1 Introduction

Power electronics has developed quickly since the advent of power semiconductor

devices in the 1950s and has become a significant technology for variants of applications in

the industrial, commercial, residential, aerospace, military, and utility areas [218]. The model,

design, and analysis of power electronic circuit (PEC) are the fundamental and significant

research areas in the power electronics. PEC always consists of a number of components such

as resistors, capacitors, and inductors which have to be optimized design in order to obtain

good circuit performance. Suitable components design and control parameters tuning of the

PEC often challenge the engineers because they may require systematic procedure.

Traditional approaches include the state-space average method [219][220], current injected

equivalent circuit method [221][222], sampled-data modeling method [223], and state-plane

analysis method [224], etc. However, these approaches are usually only applicable for

specific circuits and require comprehensive knowledge on the circuit operation. Moreover, as

these approaches are based on small-signal models, circuit designers would sometimes find it

difficult to predict precisely the circuit responses under large-signal conditions.

With the rapid development of the power electronics technology and the growing

complexity of PEC, automatic design and optimization of PEC have become great need.

Since the 1970s, variants of optimization approaches such as heuristic method [225],

knowledge based method [226], gradient descent or hill-climbing method [227], and

simulated annealing method [228], have been proposed for analog circuit design automation.

However, these approaches are very sensitive to the initial solution. Moreover, they might be

inefficient enough to search globally and are subjective to be trapped into local optima when

the problems are complex. Therefore, the obtained values for the circuit components may be

sub-optimal, leading to low satisfaction when used in practical applications.

Since EA did not need accurate mathematical model, it is an important tool to solve the

PEC design problem. Zhang et al. [229] proposed the fitness function to to evaluate the

performance of PEC, and used GA to optimize the circuit in 2001. The fitness function in

[229] are adopted further, and ACO [230] and PSO[231] have been successfully applied to


104

optimize the component values of PEC. The research work indicates the good performance of

the EAs on PEC optimization design. However, there still exist disadvantages in the GA and

ACO approaches that they have to consume a lot of computation before obtaining the good

component values because of the slow convergence speed. In this chapter, an effective and

efficient particle swarm optimization (PSO) algorithm, named orthogonal learning PSO

(OLPSO) is adopted to optimize PEC because of its faster optimization speed and stronger

global search ability. Moreover, previous studies using GA and ACO approaches always

optimize the circuit components within careful pre-defined search ranges which are

determined by expert designers. For example, the search range for some resistors is from

470Ω to 47kΩ and the search range for some capacitors is from 0.33μF to 33μF. However,

such search ranges are difficult to be defined for different components in different PECs

when used in practical applications. Therefore it is of practical value to develop an effective

and efficient approach that can optimize the components with free component configurations

that are set to commonly used ranges. Based on the problem, this chapter proposed a PEC

optimization model with “free search range”. In this model, there is no need to predefine

the range of the circuit components by the experts, but only provide a free search range

according to the practical industrial application and market supply. For example, the search

range of all resistors can be 100Ω to 100 kΩ, and all capacitors can be valued from 0.33μF to

33μF [232].

This optimization model fits more the pratical application requirement, but also creates

changelles to the algorithms. In order to effectively solve the PEC optimzaiton design

problem with free search range, this chapter will apply the OLPSO proposed in Chapter 3 to

solve the problem. The successful application of OLPSO on PEC optimization problem

verify the expansibility of the OLPSO algorithm on engineering area. The important

contributions and innovations mainly include 3 points as follow:

1) Point out that the search ranges which are determined by expert designers in traditional

PEC optimization model causes the limitation of its practical application, and hence proposed

a PEC optimization model with “free search range” defined according to the practical

industrial application and market supply, which meets the practical application requirements.

2) Design a rapid global optimization algorithm based OLPSO to efficiently solve the PEC

optimization problem with “free search range”, and the algorithm is compared with GA, PSO,

other CLPSO algorithm with good performance, and JADE to verify the effectiveness and

efficiency of the CLPSO.


105

3) Extend the PSO to the engineering application problem, PEC optimization design, and test

and verify the performance of the algorithm in continuous and discrete search space, which

ensures that the obtained optimized components are avalibale from the market and the

algorithm can solve the problem efficiently.

5.2 Power Electronic Circuit

PEC is a circuit that contains a number of components such as resistors, capacitors, and

inductors. Fig. 5-1 shows the basic block diagram of a PEC. In this PEC, the circuit can be

decoupled into two parts where the first part named the power conversion stage (PCS) and

the second part is the feedback network (FN).

],,,[ 21 FRF RRRR =

],,,[ 21 FIF IIII =],,,[ 21 FCF CCCC =

],,,[ 21 FRF RRRR =

],,,[ 21 FIF IIII =],,,[ 21 FCF CCCC =

],,,[ 21 PRP RRRR =],,,[ 21 PIP IIII =],,,[ 21 PCP CCCC =

],,,[ 21 PRP RRRR =],,,[ 21 PIP IIII =],,,[ 21 PCP CCCC =

Fig. 5-1 A block diagram of PEC.

The function of PCS is to transfers the power from the input source vin to the output load

RL. It consists of RP resistors, IP inductors, and CP capacitors. On the other hand, FN is the

control part that consists of RF resistors, IF inductors, and CF capacitors. There is a signal

conditioner H in the FN circuit to convert the PCS output voltage vo into a suitable form 'ov

which is used to compared with the reference voltage vref. Their difference vd is then sent to

an error amplifier in order to obtain a output ve. This output is combined with the feedback

signals Wp from the PCS part to give an output control voltage vcon. Then vcon is modulated by

a pulse-width modulator to give a feedback voltage vg to the PCS part.


106

In order to optimize the component values of PEC, the components are coded as the

variables and are optimized through the optimization process. Although the components of

PCS and FN can be optimized together in a single process, it is computationally intensive

since the number of variables is considerably large. Moreover, as the interactions between the

two parts in the optimization are relatively low during the training process, the components in

the two parts can be optimized separately [229].Therefore, this chapter adopts the technique

to consider the components of PCS and FN separately. This decoupled technique is not only

effective to reduce the computational effort, but also is helpful for obtaining better

component values. However, as the PCS part is always with static characteristics and the

component values are relative stable, the components in PCS are not optimized in this chapter,

but the components in the FN part which are crucial to the circuit performance are optimized

by OLPSO.

5.3 OLPSO FOR PEC

5.3.1 Particle Representation

Using OLPSO to solve the PEC, the components in the FN part can be represented with

the use of an vectors X(FN). Specifically, the representation of each particle for optimizing

the FN components is coded as：

][)( FFF CIRFNX = (5-1)

where ],,,[ 21 FRF RRRR = are the resistors, ],,,[ 21 FIF IIII = are the inductors, and

],,,[ 21 FCF CCCC = are the capacitors.

5.3.2 Fitness Function

The fitness function definition for FN is according to the proposals in [229] whose main

considerations includes reducing the settling time and controlling the overshoot. The fitness

function is described as:

)()],,(),,(),,([)( 4, ,

321

max_

min_

max_

min_

XFXvRFXvRFXvRFXL

LLL

in

ininin

R

RRR

v

vvvinLinLinLFN +++=Φ ∑ ∑

= =δ δ (5-2)

where RL_min and RL_max, vin_min and vin_max are the minimal and maximal values of RL and

vin, respectively. δRL and δvin are the step length in varying the values of RL and vin.


107

The F1, F2, F3, and F4 are the four objective functions for the FN as designed in [229].

Specifically, F1 is to measure the steady-state error of the output voltage vo.. Define a

cumulative variance equation E to evaluate the proximity of output voltage vo and the

reference voltage vref in NS=⎣(vin_max –vin_min)/δvin ⎦ simulation points as：

2

1[ ( ) ]

sN

o refm

E v m v=

= −∑ (5-3)

If the value of E2 is smaller, then the stable state error is small, so the value F1 of

obtained by equation (5-4) is larger：

2/1 1

E KF K e−= (5-4)

Where is K1 the reachable maximize value of F1, and K2 用 is used to adjust the

sensitivity of F1 value to E.

In addition, F2 is to measure the transient response of vd, including the maximum

overshoot and undershoot, and the settling time; F3 is to control the steady-state ripple

voltage on the output vo; F4 is to measure the dynamic behaviors during the large-signal

change. For more details of the fitness function definitions, refer to [229]. It should be note

that, the larger value of F1、F2、F3 和 F4 represent better performance of the circuit. Thus,

fitness function defined in (5-2) is a maximization problem.

5.4 Experiments and comparisons

5.4.1 Circuit Configurations

In this section, the OLPSO algorithm is applied to the PEC design and optimization

problem. A buck regulator with overcurrent protection as shown in Fig. 5-2 is used as

simulation case. The performance of OLPSO in optimizing the PEC is evaluated and

compared with both the GA approach [229] and the PSO approach [231]. Since the PCS part

is always with static characteristics and the components L and C are relatively stable [229]

the components in PCS are not optimized in this chapter, but the values are set as 200μH and

1000μF for L and C, respectively, according the proposals in [229] and by the considerations

of available component values in industry. For FN, all component values are required to be

optimized. That is, the components R1, R2, RC3, R4, C2, C3, and C4 in the FN part are

optimized by OLPSO and the fitness function is as (5-2) and the code are as shown in (5-1).


108

As mentioned in introduction in this chapter, the components’ search ranges are difficult

to be defined for different PECs. In this chapter, we set the components search ranges freely

according to commonly used ranges. That is, the search range for resistors R1, R2, RC3, and R4

are all set to be [100Ω, 100kΩ] and the search range for capacitors C2, C3, and C4 are all set

to be [0.1μF, 100μF].

Fig. 5-2 Circuit schematics of the buck regulator with overcurrent protection.

Moreover, the required specifications of the whole PEC are listed as follows：

Input voltage range vin: 20 ~ 40 V

Output load range RL:：5 ~ 10 Ω

Nominal output voltage: 5 V±1%

Switching frequency：20 kHz

Maximum settling time：20 ms

5.4.2 Algorithm Configurations

The performance of OLPSO in optimizing the PEC is evaluated and compared with both

the GA approach proposed in [229] and the PSO approach in [231], and the CLPSO [59] and

JADE [177] which perform well in function optimization area.

The parameters of GA and PSO are set according to the configurations in their

references. The crossover and mutation probabilities of the GA approach are set the same as

in [229] where px=0.85 and pm=0.25. According to the parameter suggestion of PSO in [231],


109

the inertia weight ω in PSO and OLPSO linearly decreases from 0.9 to 0.4, while the

acceleration coefficients c1 and c2 in PSO and the acceleration coefficient c in OLPSO are all

set to be 2.0. Moreover, is also 2.0 and the parameter G is set to be 5. The parameters of

CLPSO and JADE are set as the configurations in [59] and [177], respectively. For the

population size and the maximal generation number, they are set to be 30 and 500

respectively in both GA and PSO as proposed in [229] and [231], respectively. The

population size for CLPSO and JADE are set as N=40 and N=30 as suggested in [59] and

[177], respectively. However, in order to make a fair comparison, all the algorithms use the

same maximal fitness evaluations (FEs) of 1.5×104 (the value of 30×500) as the termination

criterion. As the evaluation of the fitness function is usually the most expensive

computational part in the optimization of PEC, the execution time of different algorithms will

be almost the same if they use the same number of FEs.

It should be note that, according to the experimental results of OLPSO in Chapter 3 in

this chapter, the OLPSO-L on local version performs better than the global version OLPSO-G,

so we adopt OLPSO-L to solve the PEC optimzaiton design problem, and use the name of

OLPSO in the following chapter. The population size of OLPSO is set as N=40. In order to

make the comparisons in a statistical sense, the experiment is carried out 30 times

independently with each approach and the average results are used for comparison.

5.4.3 Comparisons on Fitness Quality

The PEC optimzaiton is a maximization problem. The results of GA, PSO, CLPSO,

JADE, and OLPSO are compared in Table 5-1 where the “Mean” stands for the average

fitness value of the 30 independent runs and “Std. Dev” is the standard deviation. Moreover,

the “Best” fitness values among the 30 runs are given and compared in the table. It can be

observed from the table that OLPSO achieves best results when measured by the mean fitness

value. Therefore, OLPSO is capacity to obtain good solutions consistently. The Wilcoxon test

further confirms that OLPSO outperforms other algorithms significantly and the results

obtained by OLPSO are remarkably better. Moreover, OLPSO can obtain the highest “Best”

fitness solution among the three approaches, indicating the strong global search ability of

OLPSO. The obtained component values in the “Best” fitness solution optimized by different

approaches are presented in Table 5-2. Moreover, OLPSO also obtain the highest “Worst”

and “Medium” solutions, which indicates that OLPSO can obtain high quality solutons in

most cases.


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It is should be reminded that these results in Tbale 5-1 are obtained in the new

configured large search space. The total failure of GA indicates that this approach is not

efficient enough to make sufficient search in the large space to find the good solution, even

though it is promising in the carefully pre-defined search range [229]. Moreover, the large

search space challenges the search ability of traditional PSO, CLPSO, and JADE. The

OLPSO algorithm is still promising and its results are demonstrated to be much better than

the others. Table 5-1 Experimental Result Comparisons of Different Approaches

Algorithm Mean Std Wilcoxon Test Best Medium Worst Mean

FEs Success

GA 109.636 9.0058 Z=6.64578† 127.494 109.502 97.191 × 0 PSO 152.110 21.2900 Z=5.31537† 192.304 137.879 137.699 3817 8

CLPSO 155.902 13.9257 Z=5.31515† 191.638 149.147 138.919 8588 14 JADE 135.711 25.1273 Z=6.02482† 183.208 135.361 95.973 8681 8

OLPSO 183.749 14.1892 NA 192.962 192.425 137.924 3675 28 †The difference is significant at α=0.05 by Wilcoxon test.

Mean FEs indictes the mean FEs needed to find a solution with fitness value larger than 150. Success indicates the number of runs that the algorithm finds a solution whose fitness value is larger than 150.

Table 5-2 Optimized Component Values in the Best Run with Different Approaches

Components GA PSO CLPSO JADE OLPSO R1 356.099 Ω 100 Ω 100 Ω 100 Ω 100 Ω R2 60.4418 kΩ 71.7442 kΩ 36.4732 kΩ 82.223 kΩ 13.1202 kΩ RC3 98.6189 kΩ 831.532 Ω 960.373 Ω 136.366 Ω 1.04713 kΩ R4 2.07867 kΩ 11.4945 kΩ 115.497 Ω 100 Ω 11.1206 kΩ C2 19.6276 μF 0.1 μF 0.1 μF 0.1 μF 0.1 μF C3 28.0941 μF 1.72671 μF 1.47557 μF 6.5245 μF 1.11032 μF C4 3.38356 μF 0.1 μF 9.85044 μF 14.1778 μF 0.1 μF

Fitness value 127.494 192.304 191.638 183.208 192.962

5.4.4 Comparisons on Optimization Speed and reliability

Besides the high solution quality of OLPSO, the fast optimization speed and strong

algorithm reliability of OLPSO are also supported by the comparisons in Table 5-1. By

giving an acceptable fitness value of 150, OLPSO can successfully obtain final solutions with

fitness values larger than 150 in 28 out of the 30 runs whose figure is the twice of that of

CLPSO, whilst PSO and JADE can only succeeds in 8 runs and GA totally fails in obtaining

solutions with fitness values larger than 150. Thus, OLPSO is the most reliable algorithm to

solve PEC optimization problem. Moreover, among the successful runs in each algorithm, the

mean FEs needed to reach the acceptable value of 150 in Table 5-1 further shows that

OLPSO is the fastest algorithm among the three contenders.


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The mean convergence characteristics of different approaches are plotted in Fig. 5-3.

The curves show that both GA and JADE fall into a poor local optimum quite early, whilst

OLPSO is able to obtain very high fitness in early state and to improve the fitness value

steadily for a long time. Although PSO and CLPSO can find good solution in a certain range,

but the results are worse compared with OLPSO. OLPSO has strong global search ability to

avoid local optima and has significantly improved the fitness value. Moreover, the curves

indicate that OLPSO is faster than the other algorithms to optimize the component values.

That is, when a fixed fitness is given, OLPSO is observed to use much less FEs to obtain this

specific value than the GA or PSO algorithm. OLPSO is a effective and efficient algorithm to

solve the PEC optimization problem.

0.0 3.0x103 6.0x103 9.0x103 1.2x104 1.5x104100

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140

150

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200

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n fit

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GA PSO CLPSO JADE OLPSO

Fig. 5-3 Mean convergence characteristics of different approaches in optimizing PEC.

5.4.5 Comparisons on Simulation Results

In the simulations, the component values of the PEC are set as the medium optimized

results in 30 runs obtained by GA, PSO, CLPSO, JADE, and OLPSO respectively, as given

in Table 5-2. In the simulation results comparison, Fig. 5-4 gives the results of voltage and

Fig. 5-5 gives the results of current.

The simulation lasts for 90 milliseconds (ms). The input voltage vin is 20 V and the

output load RL is 5 Ω. The simulated startup transients can be compared in the first 30 ms of

the figures. It is observed that the circuit with OLPSO-optimized component values has better

performance, giving faster settling time. The buck with component values optimized by

OLPSO uses only about 5 ms to reach the steady state, while the one with component values

optimized by GA uses about 10 ms. A high voltage impulse appears during startup in the

JADE-optimized circuit, which decrease the application of the circuit. Moreover, the output


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ripple voltage of the OLPSO-optimized circuit is less than 1%, satisfying the required

specification very well

Fig. 5-4 and Fig. 5-5 also show the simulated transient responses under large signal

disturbances. On the 30 ms, when the regulator is in steady state, the input voltage is

suddenly changed from 20 V to 40 V, with the load still fixed as 5 Ω. As the responses to this

change, the output voltage vo, the control voltage vcon, and the inductor current iL are all

disturbed. However, the circuit optimized by OLPSO has much smaller disturbance and

shorter response time than the one optimized by GA, PSO, CLPSO, and JADE (2ms vs 12ms,

5ms, 10ms, and 3ms), confirming the advantages of the OLPSO algorithm. In addition, the

transient overshoot of the output velatage in GA-optimized and PSO-optimized circuits are

too large that they are unsuitable for the practical application of the circuit.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 900

1

2

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vo

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0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 900123456789

vcon

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0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 900

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(e) OLPSO Fig. 5-4 Simulated voltage responses from 0 ms to 90 ms.


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(a) GA (b) PSO

(c) CLPSO (d) JADE

(e) OLPSO

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 900.0

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Fig. 5-5 Simulated current responses from 0 ms to 90 ms.

Similar tests on load disturbances are also studied when the system has reverted a steady

state with vin equals 40 V and RL equals 5 Ω. In this disturbance, RL is suddenly changed from

5 Ω to 10 Ω on the 60 ms, with the vin is still fixed as 40 V. The simulation results in the

figures also show that the OLPSO-optimized circuit has a smaller disturbance response to the

change and a shorter time to revert the steady state. Therefore, the OLPSO algorithm can

optimize the circuit component values to make the circuit exhibit better dynamic

performance.


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5.4.6 Comparisons on Discrete Search Space

The comparisons on fitness quality, optimization speed, and simulation results have

demonstrated that OLPSO performs much better than both the GA, PSO, CLPSO and JADE

approach. However, all these results are based on the continuous search space and therefore

the optimized component values are sometimes not readily available in manufactory, but

require post-fabrication. For example, the results in Table 5-2 show that the optimized value

for the resistor R2 is 13.1202 kΩ and the optimized value for the capacitor C3 is 1.11032 μF.

These values are needed to be composed by connecting several resistors or capacitors in

series and/or parallel, and therefore make it not easy to use in practical applications. As the

resistors and capacitors are always manufactured with discrete values [232], we will test the

search abilities of different algorithms in obtaining the optimized component values in the

discrete search space. Table 5-3 Experimental Result Comparisons on Discrete Search Space

Algorithm Mean Std Wilcoxon Test Best Medium Worst Mean

FEs Success

GA 104.229 7.0487 Z=6.64846† 128.089 102.851 96.9369 × 0 PSO 156.287 26.8560 Z=3.24948† 191.842 146.778 111.801 2063 13

CLPSO 155.919 12.8958 Z=4.28188† 188.885 160.215 133.374 10868 18 JADE 144.281 19.5665 Z=5.11015† 187.7 142.572 95.9546 8455 7

OLPSO 176.550 19.5284 NA 192.199 182.996 134.021 4072 25 †The difference is significant at α=0.05 by Wilcoxon test.

Mean FEs indictes the mean FEs needed to find a solution with fitness value larger than 150. Success indicates the number of runs that the algorithm finds a solution whose fitness value is larger than 150

Table 5-4 Optimized Component Values in the Best Run of Different Approaches in Discrete Search Space

Components GA PSO CLPSO JADE OLPSO R1 200 Ω 100 Ω 100 Ω 100 Ω 100 Ω R2 91 kΩ 30 kΩ 43 kΩ 56 kΩ 10 kΩ RC3 100 kΩ 620 Ω 5.1 k Ω 510 Ω 1.1 kΩ R4 4.7 kΩ 100 Ω 11 k Ω 100 Ω 100 Ω C2 43 μF 0.1 μF 0.1 μF 0.1 μF 0.1 μF C3 36 μF 2.2 μF 1.1 μF 2.7 μF 0.91 μF C4 1.8 μF 12 μF 0.1 μF 11 μF 11 μF

Fitness Value 128.089 191.842 188.885 187.7 192.199

In the following experiments in this sub-section, the optimization algorithms work the

same as they are searching in a continuous search space except that when evaluating the

fitness value of an individual, the variables are first rounded to the nearest readily available

value. For example, in the E24 series of resistor and capacitor, the resistor values are among

{…, 180Ω, 200Ω, 220Ω, …, 12kΩ, 13kΩ, 15kΩ, …} and the capacitor values are among

{…, 0.33μF, 0.36μF, 0.39μF, …, 1.1μF, 1.2μF, 1.3μF, …} [232]. Therefore, if the algorithm


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finds a resistor value of 13.1202kΩ, it will be rounded to 13kΩ, and if the capacitor value is

1.11032μF, it will be rounded to 1.1μF.

We apply such a strategy to the GA, PSO, CLPSO, JADE, and OLPSO algorithms to

test their search abilities in discrete space whose ranges are still the same as free search range

given in Chapter 5.4.1. The experimental results are compared in Table 5-3 and the obtained

component values in the “Best” fitness solution optimized by different approaches are

presented in Table 5-4. The results show that GA still totally fails in the discrete search space,

both of JADE and PSO only success less than 50% (7 times and 13 times out of 30 trials,

respectively), and CLPSO successes 18 times. OLPSO performs better and successes 25

times. The mean fitness obtained by OLPSO is much better than GA, PSO, CLPSO, and

JADE, as indicated by the Wilcoxon test. Therefore, OLPSO has advantages in optimizing

the PEC not only when the search space is continuous, but also when the search space is

discrete, especially on PEC optimization problem with free search range.

5.5 Chapter Summary

This chapter presents an orthogonal learning PSO proposed in Chapter 3 for optimizing

the component values in designing PEC. The challenge of the problem is that the components

interact with each other and make the search space complex. Moreover, the PEC optimization

model is difficult to be described by accurate mathematic model. Evolutionary algorithms

such as PSO did not need any and global search ability, which makes the promising

application on PEC optimization problem. Previous studies using GA, ACO, and PSO

approaches have been reported, but the solution accuracy and convergence rate are still

needed to be improved. Moreover, the circuit components within careful pre-defined search

ranges in the studied optimization model are always determined by expert designers, which

strongly influence the performance of the algorithms on practical PEC optimization

problems.

To improve the algorithm and PEC optimization model, this chapter extended the PEC

optimization model, and proposed a free search range model to meet the requirement of the

unpredictable range of the components in practicle engineering application. However, the

complex search space of PEC will challenge the efficiency of traditional approaches and

requires optimization approach with strong global search ability. Therefore, on the other hand,

the orthogonal learning PSO (OLPSO) proposed in Chapter 3 is adopted to optimize PEC.

Combine the faster optimization speed and stronger global search ability of OLPSO and the


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free search range characteristic of PEC, this chapter solves the new PEC optimization

problem successfully and extends the application of the algorithm in engineering.

The effectiveness and efficiency of the OLPSO algorithm in optimally designing PEC

have been evaluated with the design of a buck regulator with overcurrent protection. In order

to demonstrate the advantages of the proposed OLPSO algorithm, results obtained by GA,

PSO with traditional learning strategy, CLPSO, and JADE are compared with the ones

obtained by OLPSO. The results show that OLPSO outperforms the other algorithms not only

with higher quality fitness value, but also with faster optimization speed and stronger

algorithm reliability. Moreover, simulations results on the circuits demonstrate the

advantages of the OLPSO algorithm by showing that the circuit optimized by OLPSO

exhibits both shorter startup time and short settling time in the transient responses. The

experimental results demonstrate that the machine learning aided OLPSO can not only

performs well on the benchmark functions but also on is outperforming on practical

engineering application optimization problems.

The good performance of OLPSO presented on the PEC optimization design problem

not only indicates that the design of machine learning aided PSO algorithm has significant

impact, but also shows that OLPSO is a powerful approach to solve the complex multimodal

optimization problem.

Chapter 6 Conclusion and Future Work

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6.1 Conclusion

As an emergent global optimization method, due to the simple algorithm implement,

high operation efficiency, and more rapid convergence rate compared with other traditional

evolutionary algorithm such as GA on most problems, PSO has attracted wide attention of

academia and engineering. Although PSO has made remarkable advances in recent 20 years

since 1995, researchers often ignore the worth of the larger number of data created in the

process of the algorithm when considering how to improve the algorithm or algorithm

application. Similar to other evolutionary algorithm, PSO is also a population-based and

iterative evolution optimization method. Thus, a great number of searching data and historical

data are produced in running process of PSO, and these data implies useful information such

as personal searching path, population evolution trend, population distribution, current

running state, structural feature of the found solutions, interaction in population, interaction

between population, current advantage of the algorithm, challenges met by the problems.

How to analyze and process these data and use them to aid the implement of the algorithm,

will be an important mean to improve the performance of the algorithm. ML technique gets

new knowledge and skill by computer simulating human learning behaviors to improve the

performance and have ability of acquiring useful information from a large number of data.

Therefore, this thesis focusses on innovation research into ML aided PSO and its engineering

application. It is expected to combine the ML and EC these two important area in computer

science, and make important attempts using ML aided PSO and ML assisted EC methods for

algorithm design and application.

For the ML aided PSO algorithm design and its engineering application problem, the

main work in this thesis is to use statistical analysis, orthogonal prediction technique and

ensemble learning to improve the algorithm from parameter control, operator design, and

population interaction three levels, and applied successfully the novel effective PSO

algorithm on one practical engineering application problem of power electronic circuit design.

Specific summary is as follow:

Firstly, this thesis bases on the statistical analysis technique in ML, and proposes

adaptive PSO (APSO) to improve the universality of the algorithm.


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APSO use statistical analysis technique to analyze and utilize the population data and

fitness value data created in the running process of PSO to implement the evolutionary state

estimation and partition. Finally, the algorithm adaptively controls the parameters and

strategies according to the current evolutionary state, and increases the convergence rate of

the algorithm and avoid falling into local optima. Experiments on 12 unimodal and

multimodal benchmark functions are carried out to evaluate the performance of the algorithm.

The results show that APSO is able to adaptively adjust the parameters according to the

optimization environment under different running state to get rapid convergence rate on

unimodal and multimodal problems. Meanwhile, since the elite learning strategies (ELS) in

APSO can be adaptively carried out when the algorithm has converged, and enhance the

ability of jumping out the local optima, and can converge to global optima on multimodal

problem. The experiments results and comparison analysis show that ML aided PSO APSO

has good convergence rate, strong global searching ability, and stable algorithm reliability,

and is an important and successful exploration on ML aided PSO design.

Secondly, based on orthogonal prediction technique in ML, this thesis proposes

orthogonal learning particle swarm optimization (OLPSO) to increase the rapid global

searching ability of PSO.

For the problem that the traditional learning strategies in PSO is not able to make full

use of personal and swarm experience information, inspired by the ML technique orthogonal

design and orthogonal prediction that can discover useful information and provide effective

prediction, the OLPSO proposed a new orthogonal learning (OL) method to modify the

velocity update operator of PSO. The OL method employed orthogonal combination

technique to combine the personal historical best experience and population historical best

experience to discover useful information and predict and construct a guidance vector with

best searching experience to guide the flight of the particles. The method using only one

learning vector with correct guidance direction to lead the particle to fly, can obvious “local

fluctuation” problem as well as “two steps forward, one step back ” problem. The

experiments on 16 unimodal, multimodal, rotation, and shift functions and comparison with

traditional PSO, improved PSO, and other well perform evolutionary computation methods

are taken to verify the contribution of OL strategy to the algorithm convergence rate and

solution accuracy, and demonstrate that OLPSO has rapid global searching ability, and is a

successful and significant attempt to explore the ML aided PSO algorithm design and also a

efficient tool to solve the complex multimodal global optimization problem.


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Thirdly, learning from the idea of ensemble learning in ML, this thesis proposes a

co-evolutionary multipopulation multiobjective particle swarm optimization (CMPSO) to

increase the application effect of the algorithm in multiobjective area.

CMPSO learns from idea of using the multiple classifiers combination method to

enhance classify effect in ensemble learning in ML, and adopts multiple coevolution

technique in PSO to solve the multiobjective optimization problems. Similar to the concept

that one classfier corresponds to one classification in ensemble learning, an optimization

framework of multiple population for multiple objective (MPMO) is to use multiple

population to optimize multiple objectives, and one population optimize one objective. Based

on the framework of MPMO, CMPSO proposes a new method with external archive sharing

to implement the information sharing between population and coevolution, and modifies the

velocity update operator to enhance the algorithm convergence rate and uses elite learning

strategy (ELS) in archive to avoid being trapped into local optima. The experimental results

reported on 18 MPO benchmark problems with different characteristics demonstrate that

CMPSO can find nondominant solution set uniformly distributed along the Pareto front and

performs well on the multiobjective problems. As an algorithm inspired by ML technique,

CMPSO is a successful and important exploration on ML aided PSO algorithm design.

Finally, OLPSO is applied on power electronic circuit optimization (PEC) design to

improve the PEC model, and extend the application area of PSO.

PEC design is a complex engineering practice problem. PEC consists of a great number

of capacitor, resistors, and inductors. How to set the value of these circuit components is a

critical problem in designing a high stability circuit. In traditional ways, engineers calculates

physical equations to get a the initial result on depending on their experience, and then tune

the value to revise the circuit design by trial and error method. However, this method needs

professional knowledge and difficult to solve the increasingly complex and lacking of

accurate mathematical model circuit optimization problem. The experiments show that GA

can not obtain feasible solution on this new PEC optimization model. Therefore, this chapter

combines the global searching ability and “free searching range” characteristic of PEC

optimization to address the new PEC optimization problem. The good performance on PEC

design optimization of OLPSO illustrates that it is significant and effective to improve PSO

with the assist of ML technique and OLPSO is an efficient tool to solve the complex

multimodal optimization problem.

In summary, this thesis tracks the ML aided PSO design and its engineering application

problems, introduces the technique and idea of ML such as statistical analysis, orthogonal


120

prediction, and ensemble learning into PSO to develop innovation work from adaptive

control, orthogonal design of operator update, and multiple population coevolution

interaction three levels. Three proposed ML aided PSO are tested on benchmark functions to

verify their effectiveness and efficiency. Furthermore, this thesis applies the proposed

OLPSO to an engineering application problem as an example to present the availability of

algorithm.

Thus, as illustrated in Fig. 6-1, this thesis discovers problems in the process of algorithm

application, and then proposed three ML aided algorithm to improve the performance, and

finilly verify the effectiveness and efficiency of the proposed improved algorithm. The whole

procedure is a research and application process going from practice and return back to

practice.

Fig. 6-1 The summury.

6.2 Future work

This thesis develops a serial of innovation research on the ML aided PSO algorithm

design and its engineering application. The research work in this thesis is an important

attempt to combine machine learning and evolutionary computation these two important

research part in computer science. Based on the research results in this thesis, our future work

will be developed from several aspects:


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6.2.1 More ML techniques and EC algorithms

This thesis focuses on improving a typical EC algorithm, PSO, and its application

development. Chapter 2, 3, and 4 use statistical analysis, orthogonal design and prediction,

and ensemble learning techniques, respectively, to aid PSO to improve performance. Actually,

there are still many powerful technique, such as cluster analysis, support vector machine, and

deep learning, can be introduced into the PSO design. In future, we will make more

development work by employing more ML techniques. Meanwhile, these ML techniques not

only performs well on PSO algorithm, but also can improve other algorithms, for example,

relatively new algorithms, differential evolution algorithm [233][234] and brain storming

algorithm [235]. Therefore, our future work will include that introducing more ML

techniques into more EC algorithm to aid the algorithms to improve their performance and

extend their application area.

6.2.2 Dynamic ML aided EC algorithm

With the increasingly complexity of the engineering application problems, dynamic

environment will be an important challenges in optimization algorithm. The problem variants

and optimized objective change with the time in dynamic optimization environment, but the

changes always relate to the current or the past state. As we know, ML can learn and

summary from past and present experience, and determine and predict the future changes.

Therefore, one important part of our future work is to design ML aided EC algorithms to get

good performance on the dynamic optimization problems.

6.2.3 Distributed ML aided EC algorithm

Many current research works on EC algorithms are based on centralized computation

framework. However, with the development and the increasing complexity of the engineering

practice problems, especially problems in internet of things, cloud computing, big data and

other emerging areas, the optimized objects are often distributed being, for example, sensor

nodes distributed in internet of things, distributed compute resource and user requirements in

cloud computing, multi-source heterogeneous data from different level in different regions in

big data, these problems needs distributed optimization. How to use ML techniques to design


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distributed EC algorithms to improve the performance and application effect will be one

important work in future.

6.2.4 More Engineering Optimizaiton Practice Test

This thesis applies PSO algorithms (especially OLPSO) on power electronic circuit

design, a typical engineering practice optimization problem, to verify the efficient of the

algorithm. However, the newly presented problems in engineering practice still need EC

algorithms to further solve. So, one part of our future work is to use ML technique to analyze

the characteristics of these engineering practice problems, combine the characteristics to

propose well-focused solutions and improve the performance. The future engineering practice

optimization will focus on the emerging problems in internet of thing, cloud computing, and

big data, such as cover optimization in sensor network, positioning and tracking optimization,

resource scheduling, user intelligent management, modeling optimization and intelligent

management in big data. The experiments on more engineering practice optimization

problems, not only provide effective and efficient solutions to these engineering optimization

problems, but also extend the application area of the algorithm.

References

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