Chaos, Solitons and Fractals 35 (2008) 888–894

www.elsevier.com/locate/chaos

Chaotic annealing with hypothesis test for functionoptimization in noisy environments

Hui Pan, Ling Wang, Bo Liu *

Department of Automation, Tsinghua University, Beijing 100084, PR China

Accepted 26 May 2006

Communicated by Prof. Ji-Huan He

Abstract

As a special mechanism to avoid being trapped in local minimum, the ergodicity property of chaos has been used as anovel searching technique for optimization problems, but there is no research work on chaos for optimization in noisyenvironments. In this paper, the performance of chaotic annealing (CA) for uncertain function optimization is investi-gated, and a new hybrid approach (namely CAHT) that combines CA and hypothesis test (HT) is proposed. In CAHT,the merits of CA are applied for well exploration and exploitation in searching space, and solution quality can be iden-tified reliably by hypothesis test to reduce the repeated search to some extent and to reasonably estimate performancefor solution. Simulation results and comparisons show that, chaos is helpful to improve the performance of SA foruncertain function optimization, and CAHT can further improve the searching efficiency, quality and robustness.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Chaos is a bounded unstable dynamic behavior, which exhibits sensitive dependence on initial conditions and includesinfinite unstable periodic motions. In recent years, chaos has gained increasing interests for control, synchronization andoptimization [1,2] in the fields of physics, chemistry, biology and engineering. As a simple mechanism to avoid beingtrapped in local optima, chaos with the ergodicity property has been a novel searching technique [2]. So far, chaos-basedsearching algorithms have aroused intense interests and gained wide applications for both combinatorial and functionoptimization problems [2]. However, considering the fact that simple chaotic search often needs a large number of iter-ations to reach the global optima and is sensitive to the initial conditions, chaos-based hybrid optimization technologyhas been a hot topic recently [2]. For instance, chaotic annealing (CA) was proposed in [3], whose performances are betterthan those of the standard SA in terms of solution quality and searching efficiency. In [4], a hybrid particle swarm opti-mization (PSO) algorithm was proposed by incorporating chaos into PSO to achieve better searching performances.

To the best of our knowledge, there is no published work on chaotic search for uncertain optimization. However,many real-world optimization problems include uncertainty, which has to be taken into account. Since the uncertainty

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.05.070

* Corresponding author. Tel.: +86 10 62783125; fax: +86 10 62786911.E-mail address: liub01@mails.tsinghua.edu.cn (B. Liu).

H. Pan et al. / Chaos, Solitons and Fractals 35 (2008) 888–894 889

is often structureless, it usually depends on simulation to evaluate performance of solution. Meanwhile, the search spaceis often huge, and there are many local optima. Therefore, it is very hard to obtain globally optimal solution. Currently,the study on uncertain optimization has been a hot topic in the international academic fields [5], especially the study ondesigning effective and robust algorithms. In this paper, the performance of CA [3] for uncertain function optimizationis investigated, and a new hybrid approach (namely CAHT) that combines CA and hypothesis test (HT) is proposed.Based on simulation results, it is demonstrated that the proposed algorithm is of good searching efficiency, quality androbustness.

2. CA for deterministic optimization

Consider the following deterministic function optimization problem:

min f ðX Þ; X ¼ ½x1; � � � ; xn�s:t: xi 2 ½ai; bi�; i ¼ 1; 2; . . . ; n; ð1Þ

where f is the objective function, and X is the decision vector consisting of n variables.Simulated annealing (SA) [6] is an effective optimization algorithm motivated from an analogy between the simula-

tion of the annealing of solid and the strategy of solving combinatorial optimization problems. SA provides a mecha-nism to probabilistically escape from local optima and the search process can be controlled by a cooling schedule. Werefer to [6] for more detail about SA.

Due to the ergodicity property, chaos can be used to enrich the searching behavior and to avoid being trapped intolocal optimum. In [3], chaotic dynamics is incorporated into SA, where the following well-known logistic equation isemployed as chaos generator:

zkþ1 ¼ l � zkð1� zkÞ; 0 6 z0 6 1; ð2Þ

where l is a control parameter, z is a variable and n = 0,1,2, . . . . Although the above equation is deterministic, it exhib-its chaotic dynamics when l = 4 and z0 62 {0,0.25,0.5,0.75,1}. The track of chaotic variable can travel ergodically overthe whole search space.

By incorporating chaotic dynamic into SA, CA was proposed in [3] for function optimization in deterministic envi-ronments. The main distinctions between CA and the standard SA lie in initialization and generator for the new solu-tion. In particular, CA employs chaotic initialization and utilizes chaotic sequences as new solution generator. Due tothe hybridization of chaotic search and SA-based search, CA outperforms SA in terms of searching quality and robust-ness on initial solution. That is, chaos is helpful for SA to improve the performances in deterministic environments. Werefer to [3] for the detail procedure of CA.

3. CA for uncertain optimization

Generally, uncertain function optimization problems can be described as follows:

min JðX Þ ¼ E½LðX ; nÞ�; X ¼ ½x1; . . . ; xn�s:t: xi ¼ ½ai; bi�; i ¼ 1; 2; . . . ; n: ð3Þ

where X is the decision vector consisting of n variables, n denotes the noise, L(X,n) and J(X) denote the sample perfor-mance and its expected value respectively.

Ideally, it expects the optimization algorithm to work on the expected value J(X) while not be misled by the noise.Since the expected value J(X) cannot be estimated precisely with limited evaluations, in practice multiple evaluations areused to calculate the following mean sum of a number of random samples L(X,n) as a replacement for J(X).

LðX Þ ¼ 1

N

XN

i¼1

LðX ; niÞ: ð4Þ

where N is sample size or evaluation number.For the above uncertain optimization problem, it may be solved by using CA where LðX Þ is used as objective func-

tion. The question is that how about the optimization performances of CA for such kind of uncertain problem? Can CAobtain the theoretically optimal solution? How about the robustness of CA? And, is chaos still helpful to SA to improvethe performances? In addition, CA only stresses the searching process in solution space, is it necessary to consider the

890 H. Pan et al. / Chaos, Solitons and Fractals 35 (2008) 888–894

evaluation of solution to achieve better searching performances? In the next section a hybrid approach combining CAand hypothesis test will be proposed.

4. CAHT for uncertain optimization

4.1. Hypothesis test

Hypothesis test (HT) is an important statistical method that is used to make test for predefined hypothesis usingexperiment data [7]. To perform HT for two different solutions when solving uncertain optimization problems, it oftenneeds multiple independent evaluations to provide suitable performance estimation for decision solutions. If ni indepen-dent simulations are carried out for solution Xi, then its unbiased estimated mean value J i and variance s2

i can be cal-culated as follows:

J i ¼ LðX iÞ ¼Xni

j¼1

LðX i; nÞ=ni; ð5Þ

s2i ¼

Xni

j¼1

½LðX i; nÞ � J i�2=ðni � 1Þ: ð6Þ

Considering two different solutions X1 and X2, whose estimated performances bJ ðX 1Þ and bJ ðX 2Þ are two independentrandom variables. According to the law of large number and central limit theorem, the estimation bJ ðX iÞ subjects toNðJ i; s2

i =niÞ when ni approaches to 1. Suppose bJ ðX 1Þ � Nðl1; r21Þ and bJ ðX 2Þ � Nðl2; r

22Þ, where the unbiased estima-

tion values of l1, l2 and s21, s2

2 are given by Eqs. (5) and (6), and let the null hypothesis H0 be ‘‘l1 = l2’’ and the alter-native hypothesis H1 be ‘‘l1 5 l2’’.

If r21 ¼ r2

2 ¼ r2 and r2 is unknown, then the critical region of H0 is described as follows:

jJ 1 � J 2jP ta=2ðn1 þ n2 � 2Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn1 þ n2Þ=ðn1n2Þ

p�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ðn1 � 1Þs2

1 þ ðn2 � 1Þs22�=ðn1 þ n2 � 2Þ

q¼ s: ð7Þ

Thus, if jJ 1 � J 2j < s, i.e., the null hypothesis holds, then it can be regarded that the performances of those two solu-tions have no significant difference in statistical sense, otherwise they are significantly different. Furthermore, for uncer-tain minimization problem it is assumed that X2 is better than X1 if J 1 � J 2 P s, while X1 is better than X2 ifJ 1 � J 2 6 �s. In addition, for a specific problem it often supposes that the theoretical performance variances of all solu-tions are the same [7], so the hypothesis test can be done according to Eq. (7). Since CA is of good performances formulti-modal deterministic optimization problems, it motivates us to investigate the performance of CA for uncertainoptimization and to propose a hybrid approach combining HT with CA.

4.2. CA with HT (CAHT)

The procedure of CAHT is described in Fig. 1.It can be seen that, firstly, the CAHT inherits the fundamental framework of CA to avoid being trapped in local

optima. Secondly, aimed at the uncertainty in uncertain optimization problems, multiple independent evaluationsare used to provide reasonable performance estimation for solutions and to reduce repeated search. Meanwhile, HTis added into the algorithm to identify the quality of different solutions, and repeated search can be avoided if it isregarded that there is no significant difference between two solutions. In the following sections, we will investigatethe performance of the CAHT for function optimization in noisy environments.

5. Numerical simulations

In this paper, six deterministic optimization problems [3,6] named GP, BR, HN-3, HN-6, RA and SH are used toconstruct uncertain problems for testing. In particular, let f(X) be the deterministic function, the uncertain problem isformulated as follows:

min J ¼ E½LðX ; nÞ� ¼ E½f ðX Þ þ g � F � n�: ð8Þ

Fig. 1. Procedure of CAHT.

Table 1Simulation results in noisy environment when g = 0.1

Method Index GP BR HN3 HN6 RA SH

SA Ja(X*) 35.4580 0.4199 �3.3108 �1.5721 �1.9633 �181.4673LaðX �Þ 35.3583 0.4127 �3.3785 �1.5833 �1.9727 �194.8766k 15 32 5 0 43 /

CA Ja(X*) 33.8740 0.4138 �3.5633 �1.7695 �1.9655 �182.6321LaðX �Þ 33.8588 0.4075 �3.6863 �1.8320 �1.9800 �195.7533k 19 36 9 0 47 /

CAHT Ja(X*) 4.8688 0.3990 �3.7122 �1.9973 �1.9797 �185.0285LaðX �Þ 4.8667 0.3822 �3.8047 �2.0315 �2.0211 �195.8623k 43 43 18 3 47 /

H. Pan et al. / Chaos, Solitons and Fractals 35 (2008) 888–894 891

892 H. Pan et al. / Chaos, Solitons and Fractals 35 (2008) 888–894

where n is noise subjected to Gaussian distribution N(0,1), g denotes noise magnitude, F is a scale factor related to func-tion f(X). For the six problems, the values of F are set as 3, 0.398, 3.86, 3.32, 2 and 186, respectively.

Let g = 0.1 and N = 10 for all algorithms, and the control parameters of CAHT, SA and CA are the same as thoseused in [3]. The statistical performances of the three algorithms are listed in Table 1, where X* denotes the obtainedoptimal solution by certain algorithm in a random run. And, LaðX �Þ and Ja(X*) denote the average estimated perfor-mance and the average expected performance of all X* in 50 runs respectively, and k denotes the number of satisfactorysolutions which are close to the theoretical optimum with distances less than 0.1. Moreover, the distribution figures ofthe resulted solutions of 50 independent runs using the three algorithms for solving GP problem in noisy environmentsare illustrated in Figs. 2–4. Meanwhile, Fig. 5 shows convergence process of LðX �Þ by using CA, SA and CAHT for GPproblem in noisy environments when g = 0.1. In addition, Table 2 shows the influence of g and N on the performancesof the three algorithms when solving GP problem in noisy environments.

From the above simulation results, we can draw following conclusions.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

x 2

Fig. 2. Results of 50 runs with SA when g = 0.1.

-2 -1.5 -1 -0.5 0 0.5 1 15. 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

x 2

CSA L(x)SAL(x)

Fig. 3. Results of 50 runs with CA when g = 0.1.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

x 2

Fig. 4. Results of 50 runs with CAHT when g = 0.1.

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

140

160

180

200

Generation

L(x

)

CA

SA

CAHT

Fig. 5. Convergence process of Lðx�Þ with CA, SA and CAHT when g = 0.1.

H. Pan et al. / Chaos, Solitons and Fractals 35 (2008) 888–894 893

Firstly, the results of SA and CA for uncertain optimization problems are often not satisfactory, while it can be seenthat the results of CA are better than those of SA. So, it is concluded that chaos is still helpful to improve the perfor-mances of SA in noisy environments.

Secondly, by combining HT with CA, good solutions can be identified reliably and repeated search can be reduced,so that CAHT can achieve better optimization quality, efficiency and the probability of hitting the theoretical optima.

Thirdly, also due to the merits of HT and CA, CAHT is of good robustness on the initial conditions. From Figs. 2–4,it can be seen that the solutions by CAHT are very close to the theoretical optima, while those by SA and CA are oftenfar away from the theoretical optima.

Besides, when noise magnitude is large, the searching quality of CAHT is still acceptable by using more evaluationtimes for solution performance estimation.

Table 2Effect of g and N (GP problem is used)

Method Index g = 0.01 g = 0.05 g = 0.10 g = 0.20 g = 0.50 g = 1.00

SA (N = 10) Ja(X*) 4.9136 15.1786 35.4580 39.2280 44.9334 46.9985LaðX �Þ 4.9101 14.9924 35.3583 38.8919 44.7876 46.5982

CA (N = 10) Ja(X*) 4.9125 13.4714 33.8740 37.1633 41.5606 40.6365LaðX �Þ 4.9086 13.4693 33.8588 37.1297 41.4680 40.2983

CAHT (N = 10) Ja(X*) 4.6020 4.7706 4.8688 5.1570 5.0881 5.1765LaðX �Þ 4.6016 4.7620 4.8667 5.1354 4.9949 4.5941

SA (N = 20) Ja(X*) 4.7661 16.5803 29.4732 37.7370 41.4480 42.4980LaðX �Þ 4.7679 16.5734 29.4332 37.3971 41.0842 41.9565

CA (N = 20) Ja(X*) 4.7153 12.6133 26.8145 33.1219 40.5941 36.4901LaðX �Þ 4.7156 12.6034 26.7992 33.0366 40.2424 35.9400

CAHT (N = 20) Ja(X*) 4.5664 4.4693 5.1736 4.9820 4.9679 5.5052LaðX �Þ 4.5651 4.4633 5.1549 4.9439 4.8835 5.5153

894 H. Pan et al. / Chaos, Solitons and Fractals 35 (2008) 888–894

6. Conclusion

In this paper, a hybrid approach (CAHT) combining CA and hypothesis test was proposed for optimization in noisyenvironments. It was concluded that, chaos is helpful to improve the performance of simulated annealing for uncertainfunction optimization, and CAHT can further improve the searching efficiency, quality and robustness for functionoptimization in noisy environments. The future work is to apply the CAHT to some real uncertain optimizationproblems.

Acknowledgements

This paper is partially supported by National Natural Science Foundation of China (Grant No. 60204008, 60374060and 60574072) and National 973 Program (Grant No. 2002CB312200).

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