differential evolution with tabu list for solving nonlinear and mixed-integer nonlinear programming...

PROCESS DESIGN AND CONTROL

Differential Evolution with Tabu List for Solving Nonlinear andMixed-Integer Nonlinear Programming Problems

Mekapati Srinivas and G. P. Rangaiah*

Department of Chemical and Biomolecular Engineering, National UniVersity of Singapore,4 Engineering DriVe 4, Singapore 117576

Differential evolution (DE), a population-based direct-search algorithm, has been gaining popularity in therecent past due to its simplicity and ability to handle nonlinear, nondifferentiable, and nonconvex functions.In this study, a method, namely, differential evolution with tabu list (DETL), is described and evaluated forsolving constrained optimization problems encountered in chemical engineering. It incorporates the conceptof tabu search (TS) (i.e., avoiding revisits during the search) in DE mainly to improve its computationalefficiency. DETL is initially applied to many nonlinear programming problems (NLPs) involving 2-13variables and up to 38 constraints. It is then tested on several mixed-integer nonlinear programming problems(MINLPs) encountered in chemical engineering practice. The performance results of DETL, DE, and modifieddifferential evolution (MDE) (Babu, K. V.; Angira, R.Comput. Chem. Eng.2006, 30, 989), for both NLPsand MINLPs, are presented, and the relative performance of the three methods is discussed.

Introduction

Many process design, synthesis, control, and schedulingproblems in engineering involve formulating and solvingnonlinear programming (NLP) or mixed-integer nonlinearprogramming (MINLP) problems with constraints. Examplesfrom the chemical engineering area include heat exchangernetworks,1 pooling problem,2 utility and refrigeration systems,3

and evaporation systems.4 In general, the problem can be statedas

wherex andy are vectors representing continuous and discretevariables, respectively, andh andg are equality and inequalityconstraints, respectively. The numbers of equality constraints,inequality constraints, continuous variables, and discrete vari-ables are, respectively,m1, m2, n, and (p - n). The challengingcharacteristic of NLP/MINLP problems is the existence ofnonconvexities because of either the objective function and/orconstraints. The use of traditional local optimization techniquesfor solving these problems leads to local solutions, and hence,the study of global optimization techniques for NLP/MINLPproblems has been of immense interest in the recent past.5-7

Generally, global optimization techniques can be broadlydivided into two types: deterministic and stochastic. Several

deterministic algorithms have been proposed for the solutionof NLP/MINLP problems in the literature.5,8-10 Kocis andGrossmann8 have solved MINLP problems using the outerapproximation/equality relaxation (OA/ER) algorithm. OA/ERconsists of two phases. In phase I, nonconvexities that cut offthe global optimum are systematically identified with local andglobal tests. In phase II, a new master problem is solved tolocate the global optimum that may have been overlooked inphase I. Floudas et al.9 proposed an approach to solve NLPand MINLP problems that involves the decomposition of thevariable set into two sets: complicating and noncomplicatingvariables. The decomposition of the original problem inducesspecial structure in the resulting subproblems, and a series ofthese subproblems are solved based on the generalized Bendersdecomposition for the global optimum. Ryoo and Sahinidis10

proposed a branch-and-bound-based method for MINLP prob-lems. It is based on the solution of a sequence of convexunderestimating subproblems generated by evolutionary sub-division of the search region. Adjiman et al.5 proposed two newglobal optimization techniques for MINLP problems involvingfunctions that are twice-differentiable in continuous variables.Both the techniques are based on theR-branch-and-bound (RBB)global optimization algorithm11 for twice-differentiable NLPproblems.

Most of the above deterministic methods have a mathematicalguarantee to provide the global optimum while exploiting themathematical structure of the given problem such that theoriginal problem is decomposed into a number of subproblemsthat can be solved easily. On the other hand, stochastic methodsare problem-independent and converge to the global optimumwith probability approaching 1 as their running time goes toinfinity.12 Further, they are applicable to nonconvex and/ornoncontinuous functions.

Das et al.13 studied four different versions of simulatedannealing (SA) for scheduling serial, multiproduct batch pro-

* Author for correspondence. E-mail: [email protected]. Fax:(65) 6779 1936. Phone: (65) 6516 2187.

Minimize f(x, y)

subject to hi(x, y) ) 0, i ) 1, 2, ...,m1

gj(x, y) g 0, j ) 1, 2, ...,m2

xlk e xk e xu

k, k ) 1, 2, ...,n

ylk e yk e yu

k, k ) 1, 2, ..., (p - n)

7126 Ind. Eng. Chem. Res.2007,46, 7126-7135

10.1021/ie070007q CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 10/03/2007

cesses; of the four versions of SA studied, the Metropolisalgorithm with the Aarts and van Laarhoven annealing schedulewas found to give the best results. Salcedo14 proposed anadaptive random search method for NLP and MINLP problems.The results obtained reveal the adequacy of random searchmethods for nonconvex NLPs and MINLPs in chemicalengineering. Cardoso et al.15 proposed an SA approach for thesolution of MINLP problems. The method combines the originalMetropolis algorithm16 with the nonlinear simplex method ofNelder and Mead.17 The proposed approach is shown to bereliable and efficient, especially for larger-scale and ill-conditioned problems. Jayaraman et al.18 applied an ant-colonyframework for optimal design of batch plants and found it tobe robust to locate the optimum with<0.04% error. Yu et al.19

proposed a new algorithm combining both genetic algorithm(GA) and simulated annealing (SA) to solve large-scale systemenergy integration problems. Their results show that the newalgorithm can converge faster than either SA or GA alone andhas a higher probability of locating the global optimum.

Costa and Oliveira20 examined both GAs and evolutionarystrategies (ESs) for MINLP problems. Their results show thathighly constrained problems are challenging for ESs; but, ingeneral, they are efficient in terms of function evaluationscompared to GA and SA-based modification of simplex methodand simulated annealing (M-SIMPSA) of Cardoso et al.15 Linand Miller21 studied tabu search (TS) for the solution of NLPand MINLPs. Several constraint-handling techniques and initialvalues for key parameters of TS are also described in this work.The results demonstrate the effectiveness of TS for chemicalengineering problems. Danish et al.7 presented modified GAby combining various effective schemes (such as simulatedbinary crossover, polynomial mutation, and variable elitismoperators) proposed by several researchers. Modified GA is thenused to solve several multiproduct batch plant design problems,and results reported are comparable to or better than those inthe literature.

Lampinen22 applied differential evolution (DE) for solvingnonlinear constrained problems, and his results demonstrate theability of DE for NLPs. Recently, Angira and Babu6 haveapplied a modified differential evolution (MDE) for solvingseven problems in process synthesis and design. Their resultsshow that MDE converges faster than the original DE of Stornand Price.23 Because of the simplicity, ease of use, and fasterconvergence properties of DE compared to GA,24 the formerhas been used for many applications. Recently, we proposed ahybrid method, namely, differential evolution with tabu list(DETL) and comprehensively evaluated for unconstrained(except for bounds) global optimization problems.25 It incor-porates the concept of TS (i.e., avoiding revisits during thesearch) in DE mainly to improve its computational efficiency.DETL is inspired by our experience with both DE and TS;26,27

the results for benchmark, phase-equilibrium, and phase-stabilityproblems show that DE is more reliable than TS, whereas thelatter is more efficient than the former, primarily due to tabulist and avoiding revisits. In this study, DETL is described andevaluated for solving many NLP and MINLP problems arisingin chemical engineering practice. All these problems havemultiple minima and are, thus, suitable for testing globaloptimization methods. The performance of DETL is comparedwith that of DE and the recently proposed modified differentialevolution (MDE) of Babu and Angira.28

Differential Evolution with Tabu List (DETL)

In general, the basic DE23,29consists of three steps: genera-tion, evaluation, and selection. The generation step involves

producing the offsprings (new individuals) by mutation andcrossover operations, whereas the evaluation step calculates thefitness (objective function) value of each member of the newpopulation. The selection step allows only those individuals ofthe current generation that have better fitness value to proceedto the next generation. The process of generation, evaluation,and selection steps is repeated until either the best fitness value(global solution) is found or up to the specified maximumnumber of generations.

The concept of TS is implemented in the generation step ofDE (i.e., after crossover and mutation) to improve the compu-tational efficiency and also the diversity among the individuals.The members of the current generation are produced one at atime as in MDE28 to have a better convergence rate and tofacilitate the implementation of TS. The concept of TS isimplemented in DE by using tabu list (requires two parameters,namely, tabu radius (TR) and tabu list size (TLS)), which keepstrack of the previous evaluated points such that revisits duringthe subsequent search are avoided. After generating a newmember of the current population, it is compared to the already-evaluated points in the tabu list in terms of the Euclideandistance. If the Euclidean distance is smaller than the specifiedvalue (TR), which indicates that the new point is close to oneof the points in the tabu list, the newly generated point is rejectedconsidering that it may not give new information about theobjective function except increasing the number of functionevaluations (NFE). The rejected point is replaced by generatinga new individual until the Euclidean distance between the newpoint and to all points in the tabu list is greater than TR. Theprocess of generating new individuals, including checking theircloseness to those in the tabu list, is repeated for the specifiedpopulation size. The number of points in the tabu list is specifiedby the tabu list size (TLS) parameter, and how far the new pointsshould be from those in the tabu list is decided by the taburadius (TR) parameter. By implementing the TS concept in thegeneration step of DE, unnecessary function evaluations areavoided while maintaining diversity in the population.

The motivation and working principle of DETL are explainedby considering the modified Himmelblau (mHB)30 function,which has 4 minima (Figure 1), and the global minimum is atx ) {3, 2} with a function value of zero. The members of thepopulation using the conventional DE (points marked with “O”)and with DETL (points marked with “/”) after 2 generationsare shown in Figure 1. More diversity can be observed among

Figure 1. Contour diagram of the modified Himmelblau (mHB) function;O and/ denote the points generated by DE and DETL, respectively. GMis the global minimum, whereas LM1, LM2, and LM3 are the local minima.

Ind. Eng. Chem. Res., Vol. 46, No. 22, 20077127

the members using DETL compared to those using DE, whichconsequently helps DETL to explore the global region earliercompared to DE. The diversity can be estimated by calculatingthe Euclidean distance from each member/point to its nearestmember in the population. The mean and standard deviation ofthe Euclidean distances are 0.116 and 0.059 and 0.109 and0.079, respectively, with DE and DETL for the populationsshown in Figure 1 (i.e., after second generation). As the numberof generations increases, i.e., after the 10th generation, the meanand standard deviation of the Euclidean distances are 0.080 and0.053 and 0.118 and 0.084, respectively, for DE and DETL.Note that all the variables are normalized between 0 and 1 inthis study, and the mean and standard deviation calculated arebased on the normalized values. These results clearly show thatmore diversity can be obtained with DETL compared to DE,particularly as the number of generations increase. NFE (aver-aged over 100 trials) required by DE to locate the globalminimum for this function is 6991, whereas DETL (with TR)1 × 10-6 and TLS) 20) took 41% less NFE (4098, averagedover 100 trials). In both cases, the stopping criterion used is“convergence to the global optimum (i.e., achieving the globalminimum with an absolute error of 10-6 in the function value;and local optimization technique is not used at the end of bothDE and DETL)”. In the rest of this study, a local optimizationtechnique is implemented at the end of DETL (as well as at theend of other algorithms) to improve the accuracy of the finalsolution as well as the computational efficiency. In this study,sequential quadratic programming (SQP) with finite differencegradient is used for local optimization. Alternately, one can useany direct-search method such as the Nelder-Mead simplexmethod. However, for local optimization, direct-search methodsare known to be less efficient than the gradient-based methods.

Description of the Method.DETL begins with the selectionof values for the parameters (see Table 1 for parametervalues): population size (NP), amplification factor (A), crossoverconstant (CR), TLS, TR, maximum number of generations(Genmax), and maximum number of successive generations(Scmax) without improvement in the best function value. Thealgorithm (Figure 2) generates the initial population of size NPusing uniformly distributed random numbers to cover the entirefeasible region. The objective function and the constraints areevaluated at each individual, and the best one is captured. Theevaluated individuals/points are then sent to the tabu list, whichwill be used to ensure that the algorithm does not search againclose to these points.

The three main steps, mutation, crossover, and selection ofDE, are performed on the population during each generation/iteration. For each randomly chosen target individual (Xkk) inthe population, a mutant individual (Vkk,gen+1) is produced by

whereXr1, Xr2, andXr3 are the three randomly chosen individualsfrom the current population. The random numbersr1, r2, andr3

should be different from the running index (kk), and hence, NPshould be 4 or more for mutation. The mutation parameter oramplification factor,A, has a value between 0 and 2 and controlsthe amplification of the differential variation between tworandom individuals. In the crossover step, a trial individual isgenerated by copying some elements of the mutant individualto the target individual (Xkk) with a probability of CR. Aboundary violation check is performed to check the feasibilityof the trial individual; if any bound is violated, the trialindividual is either replaced by generating a new individual orforced to the nearest boundary (lower or upper). The trial

individual is then compared with the points in the tabu list. Ifit is near to any of the points in the tabu list, the trial individualis rejected and another point is generated through mutation andcrossover operation.

Objective function and constraints are evaluated at the trialindividual only if it is away from all the points in the tabu list.After each evaluation, the evaluated point is sent to the tabulist. For example, consider a population of size 30 and a tabulist of size 20. Once the first member of the population isgenerated (i.e., until it is not near to any of the points in thetabu list) and evaluated, it is placed in the first position of thetabu list, and the subsequent points will be placed in thecorresponding positions. Then, the 21st evaluated point replacesthe first point in the tabu list, and the subsequent points occupythe corresponding positions. Thus, the tabu list in DETL isupdated dynamically during the search to keep the latest point-(s) in the list by replacing the earliest-entered point(s). In theselection step, a greedy criterion such as the fitness (i.e.,objective function) value is used to select the better one of thetrial and target individuals. If the trial individual is selected, itreplaces the target individual in the population immediately andmay participate in the subsequent mutation and crossover

Vkk,gen+1 ) Xr1 + A(Xr2 - Xr3); kk ) 1, 2, ..., NP (1)

Figure 2. Flowchart of DETL.

7128 Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007

operations. If the target individual is better, then it remains inthe population and may participate in the subsequent mutationand crossover operations.

The above process of generation, evaluation, and selectionis repeated NP times in each generation (Figure 2). Thealgorithm runs until the stopping criterion, such as maximumnumber of generations (Genmax) or maximum number ofsuccessive generations (Scmax) without improvement in the bestfunction value, is satisfied. The best point thus-obtained overall the generations is further refined using a local optimizationtechnique and is compared to the known global minimum toassess the performance of the algorithm. Note that the globalminimum is known for all NLP and MINLP problems tested.

Handling Integers and Binary Variables. Integers andbinary variables may be encountered in chemical engineeringproblems. Within the optimization algorithm, these are alsorepresented as continuous variables and converted into integersfor evaluating the objective function and constraints. In thisstudy, integers are handled by rounding the continuous variablesto the nearest integers in contrast to the truncation as by Angiraand Babu.6 Truncation always takes the nearest lower integervalue, whereas the rounding method has equal probability tochoose between nearest lower and nearest upper integer values;the latter is unbiased and, thus, reasonable. For example, if thecontinuous variable has a value of 2.6 in one case and 2.4 inthe second case, then truncation takes the nearest lower integer(2) in both cases, whereas rounding takes the nearest higherinteger (3) in the first case and the nearest lower integer (2) inthe second case. Binary variables are also handled in the sameway as for integers except that their bounds are restricted to 0and 1.

Handling Constraint and Boundary Violations. In thisstudy, all equality constraints are eliminated by using them tosolve for suitable variables and substituting the resultingexpressions in the objective function and/or constraints. Allinequality constraints are handled using the most popular penaltyfunction method. The penalty function method converts theconstrained problem into an unconstrained one by penalizingthe infeasible solutions using penalty weights. A high value (1× 106) is used for the penalty weight for all the problems inthis study. If any constraint is violated, the absolute value ofthe constraint violation is multiplied by the penalty weight andis added to the objective function, assuming that the problemsare minimization type. If many constraints are violated, theneach absolute violation is first multiplied with the penaltyweight, and all of them are added to the objective function value.

Boundary violations are often encountered while performinga mutation operation in DE, MDE, and DETL. In this study,each and every member violating variable bounds is handledby replacing them with a new member generated randomlybetween the lower and upper bounds of variables. This approachis referred to as random generation (RG) in this work. ForMINLPs, another approach, namely, forcing to bounds (FB),in which the boundary violations are forced to the nearest lower/upper boundaries, is also used. This is because many MINLPproblems have global solutions at the bounds of decisionvariables; hence, using the FB approach will improve thereliability and efficiency at times. In addition to the RG andFB approaches, a mixed approach, in which∼50% of theboundary violations are corrected using the RG approach andthe remaining are corrected with the FB approach, is also usedfor the MINLPs tested. This is achieved by implementing theRG and FB approaches alternately.

Implementation and Evaluation

The FORTRAN code of DE is taken from the websitewww.icsi.berkeley.edu/∼storn/code.html and is modified forMDE (as stated by Babu and Angira,28 to update the populationonce the better solution is found instead of waiting for the wholepopulation as in DE) and then for DETL by including the tabulist and the tabu check. A local optimization technique is usedat the end of all three methods to improve the computationalefficiency and accuracy of the final solution. The subprogram,DNCONF in the IMSL software, is used for local optimization.For MINLP examples 7 and 8, FSQP program obtained fromAEM Design (www.aemdesign.com) is used for local optimiza-tion to avoid the unexpected fatal errors and consequent programtermination experienced with DNCONF. Both DNCONF andFSQP are based on the SQP method and use a numerical (finitedifference) gradient. Because both these programs do not handlebinary and integer variables, they are kept at the optimal solutionobtained by each global algorithm (DE, MDE, and DETL), andonly the continuous variables are refined using the localoptimization. It is reasonable to assume that the solution foundby the global method does not require refining binary and integervariables.

The methods are evaluated based on both reliability andcomputational efficiency in locating the global optimum.Reliability is measured in terms of success rate (SR) (i.e.,number of times the algorithm located the global optimum tothe specified accuracy out of 100 trials). Computational ef-ficiency is measured in terms of number of function evaluations(NFE) and CPU time required to locate the global optimum.NFE and CPU time are calculated based on the average overonly the successful trials out of 100 trials to avoid underestima-tion of the performance measures, as NFE and CPU time areexpected to be small for the failure runs. NFE includes functioncalls for evaluating the objective and for the numerical gradientin the local optimization. All the constraints are also evaluatedat each function evaluation. The computer system used inthis study is Pentium 4 (CPU 2.8 GHz, 1GB RAM) forwhich million floating point operations per second (MFlops)for the LINPACK benchmark program (available athttp://www.netlib.org/; accessed in February 2007) for a 500× 500 matrix are 282. For comparing the computationalefficiency, NFE is a better indicator for the application problemsinvolving extensive calculations for objective function evalu-ation, and CPU time is appropriate when the objective functionrequires a few calculations and contribution of the algorithmcomputations to CPU time is significant.

The stopping/termination criterion used in this study is themaximum number of generations (Genmax) or maximum numberof successive generations (Scmax) without improvement in thebest function value. These criteria are used instead of conver-gence to the global minimum as a stopping criterion because,in practical applications, the global minimum is unknown apriori and also accurately. A trial/run is said to be successfulonly if the global optimum is obtained with an absolute errorof 10-5 or less in the objective function value.

Nonlinear Programming Problems (NLPs)

The applicability of DETL for NLPs is tested by solving manytest functions and engineering design problems taken from theliterature.10,28,31The examples are carefully chosen so that theyhave multiple minima and at least one of the followingfeatures: narrow feasible region and employed in recent studies.Problems with no known multiple minima are specifically


excluded since this study focuses on global optimization forproblems with multiple minima. The numbers of binary/integer/continuous variables and equality/inequality constraints in eachexample are summarized in Table 2. The mathematical formula-tion, global optima, and local optima for each of these problemsare available in the cited references. Interested researchers canobtain an Excel file containing these details from the corre-sponding author. Some of the problems presented in theliterature10,32(e.g., examples 2, 10, and 12) have typographicalerrors; hence, the reference from which an example is obtainedis also given in Table 2.

Example 1 is the Himmelblau function with two nonlinearconstraints. The feasible region is narrow, crescent-shaped,31

and only 0.7% of the total search space defined by bounds,posing a significant challenge to optimization methods. Thisexample has one active constraint at the global minimum.Example 2 is the maximization of the net profit of a wood-pulp plant. It has 38 constraints resulting from material andenergy balances and several empirical equations. The solutionreported by Deb31 for this example, although better, violates aconstraint (62212/c17 - 110.6 - z1 g 0, with a value of-62.77118), and hence, the solution reported by Himmelblau,32

which satisfies all constraints, is used in this study. Example 3is relatively an easy problem, with the objective function andconstraints being linear or quadratic. For this example, aroundsix constraints are active at the global minimum. Example 4 isthe design of a heat exchanger network problem, which has allactive constraints at the global minimum. Michalewicz33 foundthis problem difficult to solve. Example 5 has a feasible regionof only 0.5% of the total search space and has two activeconstraints at the global minimum. Examples 6 and 7 are thetest functions with nonlinear objective function and constraints.Example 6 has two active constraints, whereas example 7 hassix active constraints at their global minima. Example 8 is areactor network design problem. It is difficult to solve becausethe local minimum value is very close to the global minimumvalue. The global solution for Example 8 is atx ) {0.771 462,0.516 997, 0.204 234, 0.388 812, 3.036 505, 5.096 052} with f) -0.388 812, whereas the local solutions are atx ) {0.390,0.390, 0.375, 0.375, 16, 0} with f ) -0.375 and atx ) {1,0.393, 0, 0.388, 0, 16} with f ) -0.388 1 (Ryoo and Sahini-dis).10 Example 9 is the maximization of the yield of a productwith respect to reaction time and temperature in a continuousstirred tank reactor. This example has two different globalsolutions with the same objective function value. Example 10is a test function with a bilinear constraint. Example 11 is thedesign of an insulated tank problem, whereas example 12 is apooling problem. For example 11, the upper bound for one ofthe variables (x4) is taken as 1 000 instead of infinity in thisstudy, which covers both local and global minima. Example 13has bilinearities, whereas examples 14 and 15 are quadraticallyconstrained problems with linear and quadratic objective func-tions. Example 16 is the design of a three-stage process systemwith recycle.

Parameter Tuning. The parameters of DE, MDE, and DETL(i.e., CR,A, NP, Genmax, Scmax, TR, and TLS) are tuned usingexamples 2, 4, 6, 8, and 9, which are found to be important indeciding the performance of these algorithms in our preliminarytesting. Tuning is carried out by varying one parameter at atime with the remaining parameters fixed at their nominal orrecent good values, in order to achieve good reliability and alsocomputational efficiency. The nominal parameter values chosenbased on the preliminary numerical experience are CR) 0.5,A ) 0.5, NP) 20, Genmax ) 50, and Scmax ) 5N (whereN isthe dimension of the problem) for both DE and MDE, and TR) 1 × 10-6 and TLS) 20, which are the additional parametersin DETL. The range of values used for each parameter whiletuning are 0.1-2 for A, 0.1-1 for CR, 20-100 for NP, 50N-500N for Genmax, and 10N-20N for Scmax, with a minimum of6 and a maximum of 10 points in the range. Although theparameters can be fine-tuned for each example as performedby Angira and Babu,6 a common set of good parameter valuesis used for each group (NLPs and MINLPs) in this study. Theywill be more useful in new applications.

All the decision variables are normalized, within the opti-mization program, between 0 and 1 for consistency, and valuesreported for tabu radius (TR) in this study are for use with thenormalized variables. The good parameter values obtained forNLPs are summarized in Table 1. The parameter values of MDEand DETL (except the additional parameters, TR and TLS,which are tuned separately) are kept the same as those obtainedfor DE for a fair comparison. Even then, if any algorithm (DE,MDE, or DETL) converges faster, it will be captured by theScmax termination criterion; when the algorithm converges faster,this criterion is satisfied earlier, resulting in smaller NFE.

Results and Discussion.All the examples are solved 100times, each time with a randomly generated initial estimate.

Table 1. Values of Parameters in DE, MDE, and DETL Used inThis Study

parameter NLPs MINLPs

DE and MDEamplification factor (A) 0.6 0.5crossover constant (CR) 0.6 0.7population size (NP) 20 20maximum number of generations (Genmax) 100N 40Nmax. number of successive generations without

improvement in the best function value (Scmax)10N 10N

DETLtabu radius (TR) N × 10-3 N × 10-2

tabu list size (TLS) 20 20

Table 2. Characteristics of the NLP and MINLP Problems Studied

example

constraints(equality+inequality)

variables(binary+

integer+ real) reference

NLPs1 2 (0+ 2) 2 (0+ 0 + 2) Deb31

2 38 (0+ 38) 5 (0+ 0 + 5) Himmelblau32

3 9 (0+ 9) 13 (0+ 0 + 13) Deb31

4 6 (0+ 6) 8 (0+ 0 + 8) Deb31

5 4 (0+ 4) 7 (0+ 0 + 7) Deb31

6 6 (0+ 6) 5 (0+ 0 + 5) Deb31

7 8 (0+ 8) 10 (0+ 0 + 10) Deb31

8a,f,g 5 (4 + 1) 6 (0+ 0 + 6) Ryoo and Sahinidis10

9a 0 2 (0+ 0 + 2) Umeda and Ichikawa34

10 1 (0+ 1) 2 (0+ 0 + 2) Sahinidis and Grossmann35

11 2 (2+ 0) 4 (0+ 0 + 4) Ryoo and Sahinidis10

12 7 (5+ 2) 10 (0+ 0 + 10) Visweswaran and Floudas36





MINLPs1b,c,d 2 (0 + 2) 2 (1+ 0 + 1) Ryoo and Sahinidis10

2b,c,d,e 2(1 + 1) 3 (1+ 0 + 2) Kocis and Grossmann37

3b,c,d 3 (0 + 3) 3(1+ 0 + 2) Floudas38

4b,c,d 9 (5 + 4) 4(1+ 0 + 3) Kocis and Grossmann39

5b,c,d,e 9 (0 + 9) 7 (3+ 0 + 4) Floudas40

6b,c 3 (0 + 3) 5 (0+ 2 + 3) Cardoso et al.15

7h 13 (0+ 13) 10 (0+ 3 + 7) Salcedo14

8h 61 (0+ 61) 22 (0+ 6 + 16) Salcedo14

a Babu and Angira.28 b Angira and Babu.6 c Costa and Oliveira.20

d Cardoso et al.15 e Salcedo.14 f Choi et al.41 g Ferreira et al.42 h Grossmannand Sargent.43


These initial estimates may or may not be feasible (i.e.,constraints may not be satisfied). The boundary violations arehandled using the RG approach for all NLPs, since most ofthese problems have global solutions inside the bounds of thedecision variables. The performance results (SR and NFE) arepresented in Table 3. The success rates of DE, MDE, and DETLare 100% for examples 1-7 (except for example 4), even thoughsome of them have narrow feasible regions (examples 1 and 5)and one of them (example 2) has many (38) constraints. Thisshows the reliability of the escaping mechanism (crossover andmutation) in these methods to escape from the local minima.For example 4, the global optimum obtained in this study atx) (579.300 6, 1 359.970 6, 5 109.970 4, 182.017 7, 295.601 1,217.982 3, 286.416 5, 395.601 1) withf ) 7 049.247 720 isslightly better than that withf ) 7 049.330 923 reported byDeb.31 In a few of the trials, both MDE and DETL convergedto a solution atx ) (579.319 9, 1 355.341 7, 5 114.594 2,182.018 8, 295.416 2, 217.981 1, 286.602 5, 395.416 2) withf) 7 049.255 921. This could be due to the flat objective functionnear the global solution, thus leading the gradient-basedoptimization technique (DNCONF used in this study) toconverge somewhat prematurely in a few trials. For examples1-3 and 5-7, NFE for DETL is less among the methods tested,which clearly shows the benefit of avoiding revisits using thetabu list in DE. For example 4, NFE for DETL is fewer thanthat for MDE and is slightly higher compared to that for DE.Although MDE is expected to converge faster compared to DE,28

NFE for the former is slightly more or comparable to that forthe latter for examples 2 and 4; however, for examples 1, 5, 6,and 7, MDE requires slightly fewer NFE compared to DE. Thisshows that the concept of early updating of the members of apopulation in MDE compared to DE may not improve compu-tational efficiency at times.

For example 8, SR of DE is slightly less (90%) than that(98%) of MDE and DETL because of the presence of compa-rable minima (i.e., the objective function value at the local andglobal minimum is-0.388 1 and-0.388 81, respectively, adifference of <0.2%) in this example. Once any algorithmreaches a comparable local minimum, exploring regions wherethe function value is better compared to that at the localminimum becomes very difficult, resulting in low SR. Forexample 9, SR of all the methods is less, at∼75%. This isbecause of the existence of a ridge (which creates numerouslocal solutions) as shown by Babu and Angira28 for this example.

For examples 8 and 9, NFE for MDE is slightly more than thatfor DE, and that for DETL is the least. SR of DE, MDE, andDETL are almost 100% for examples 10-16 except for example15. Though all these methods were able to escape from two ofthe local minima (atf ) 10.631 and atf ) 10.354) in example15, they were trapped in a constrained minimum atx ){2.605 55, 0} with f ) -86.422 205 in several trials, resultingin low SR. NFE for MDE is slightly more than that for DE,whereas DETL requires the lowest NFE for examples 10-16.

Overall, SR of DETL is high and comparable to that of DEand MDE, and its NFE is fewer than that for both DE and MDE.NFE taken by local optimization is∼1% of total NFE for bothDE and MDE. The NFE for local optimization in DETL is∼2%and is somewhat high compared to those for both DE and MDE.This could be because of the approximate final solutionsobtained at the end of DETL alone (i.e., without local optimiza-tion) due to the implementation of the tabu concept. Thepercentage reduction in NFE for DETL compared to those forDE and MDE is summarized in Table 4. The NFE for DE isslightly less compared to that for MDE for many examplestested, making the average reduction in NFE for MDE comparedto that for DE negative (-4.2%). On the other hand, the averagereductions in NFE for DETL compared to DE and MDE are33% and 35%, respectively.

The comparable performances of DE and MDE found in thisstudy are different from the findings of Babu and Angira.28 Thebest results (NFE) reported for DE and MDE in the previousstudy are, respectively, 37 810 and 31 877 for example 4, 2 074and 1 860 for example 8, and 7 996 and 7 351 for example 9;SR is either 99 or 100%. In other words, percentage reductionin NFE of MDE compared to that for DE is 16%, 10%, and8% for examples 4, 8, and 9, respectively. The other twoconstrained NLP examples tested by Babu and Angira28 haveno known local minima and, hence, are not selected in thepresent study. The differences in the results of DE and MDEby Babu and Angira28 and in the present study are due to severalreasons: (i) termination criterion (convergence to the globalminimum is used by Babu and Angira,28 whereas generaltermination criteria, Genmax and Scmax, are used in this study);(ii) use of a common set of good parameter values for a set offunctions in this study instead of for each function as by Babuand Angira;28 (iii) use of local optimization at the end of eachalgorithm in this study; and (iv) NFE being calculated basedon only the successful trials out of 100 in this study instead ofcalculating based on all 100 trials as by Babu and Angira.28

The last one is to avoid underestimating NFE since convergence

Table 3. SR and NFE of DE, MDE, and DETL for NLP Examples

DE MDE DETL

examplenumber SR NFEa SR NFEa SR NFEa

1 100 1960+ 21 100 1889+ 21 100 1108+ 242 100 9967+ 35 100 10020+ 23 100 6472+ 1353 100 26020+ 54 100 26020+ 54 100 21090+ 544 100 11755+ 1171 97 12283+ 1440 95 11999+ 12105 100 9289+ 156 100 9192+ 155 100 7119+ 1636 100 9634+ 17 100 9463+ 24 100 6980+ 337 100 13865+ 27 100 13232+ 28 100 12155+ 288 90 1682+ 28 98 1971+ 22 98 1468+ 259 75 3329+ 19 72 3534+ 18 77 1983+ 15

10 100 3526+ 4 100 3820+ 3 100 1361+ 911 99 3685+ 7 97 3601+ 8 96 1495+ 2612 100 8012+ 12 100 8650+ 11 100 4421+ 1213 99 861+ 7 99 1050+ 5 100 418+ 714 100 2399+ 10 100 2412+ 9 100 1638+ 1215 86 2100+ 14 77 2367+ 12 78 1630+ 5916 100 5419+ 8 100 5473+ 8 100 2912+ 12

a The two numbers in each of these columns are NFE required by themethod (DE, MDE, or DETL)+ NFE required for the local optimization.

Table 4. Percentage Reduction in NFE for NLP Examples

examplenumber

MDE comparedto DE

DETL comparedto DE

DETL comparedto MDE

1 3.58 42.86 40.732 -0.41 33.94 34.213 0 18.91 18.914 -6.17 -2.19 3.755 1.04 22.90 22.096 1.69 27.33 26.087 4.55 12.30 8.128 -16.55 12.69 25.099 -6.09 40.32 43.7510 -8.30 61.19 64.1611 2.25 58.80 57.8612 -7.94 44.75 48.8213 -21.54 51.04 59.7214 -0.49 31.51 31.8515 -12.54 20.10 29.0016 -0.99 46.12 46.65Average -4.24 32.66 35.05


to a local minimum in the failure runs takes fewer NFE. Ingeneral, evaluation of the methods in this study is more realisticand has wider applicability.

The CPU time (in s) of all the methods for NLPs is given inTable 5. Between DE and MDE, the latter took slightly lessCPU time compared to the former. This is because MDE usesonly one population compared to the two sets of populations inDE. Though DETL required fewer NFE, it took more CPU timecompared to both DE and MDE for most of the NLPs exceptfor a few examples (11, 13, 14, and 16). This is because of theadditional computational effort required by DETL (i) forcomparing the newly generated members in the population tothe points in the tabu list (tabu check step in Figure 2) and (ii)for generating new members that are away from those in thetabu list, compared to DE and MDE. The CPU time of DETLis less compared to DE for examples 11, 13, 14, and 16 due tothe associated fewer NFE. Though DETL took more CPU timecompared to DE and MDE for most of the NLPs tested,application problems where function evaluation takes consider-able CPU time, the additional computational effort for tabuchecks and generating new members is expected to be insig-nificant.

Mixed-Integer Nonlinear Programming Problems(MINLPs)

The applicability and efficiency of DETL is also tested forsolving several MINLP problems related to process synthesisand design studied by different authors. These problems arenonconvex optimization problems and involve binary or integervariables besides continuous variables (Table 2). Examples 1and 3 are process synthesis and process flow-sheeting problems,respectively, with nonconvexities in the first constraint. Example2 is also a process synthesis/design problem with nonlinearequality constraint. Example 4 is a two-reactor problem, wherethe selection is to be made among two candidate reactors forminimizing the cost of producing a desired product. Example5 is a process synthesis problem, whereas example 6 is a processdesign problem having multiple global solutions. Examples 7and 8 refer to the optimum design of multiproduct batch plantswith several local minima. The numbers of binary/integer/realvariables and equality/inequality constraints in each of theseexamples as well as the reference from which they are takenare summarized in Table 2. The mathematical formulation,global optima, and local optima for each of these problems areavailable in the cited references. An Excel file containing thesedetails is available from the corresponding author.

All the parameters of DE, MDE, and DETL are tuned basedon examples 2-4, which are found to be important in deciding

the performance of these algorithms in our preliminary testing.The good values obtained for the parameters in each methodfor MINLPs are also given in Table 1. They are slightly differentfrom those obtained for NLPs. This could be due to the lowernumber of variables and constraints in MINLP examples (exceptexample 8) compared to NLP examples studied. All theexamples are solved 100 times, each time with a randomlygenerated initial estimate. The three approaches, RG, FB, andmixed, for boundary violations are also tested. The performanceresults (SR and NFE) for all three methods are given in Table6.

Results and Discussion.The SR of all three methods areclose to 100% for example 1 using the RG approach. SR is notaffected using the FB approach instead of the RG approach,for both DE and DETL, but it is slightly decreased for MDE;MDE converged to the nearest local minimum (f ) 2.236 067)in 10 out of 100 trials. This could be because of the prematureconvergence associated with early updating of populationmembers in each generation compared to that of DE. Forexample 2, SR of all methods is less using the RG approach.Most of the failed runs converged to a solution atx ) 0.852 605andy ) 0 with f ) 2.557 816, which is somewhat comparableto the function value at the global minimum (2.124). By forcingthe violated members to the nearest bounds (i.e., using the FBapproach), the SR of all the methods has increased for example2. The SR of MDE is less compared to those of both DE andDETL using both RG and FB approaches for this example. Onthe other hand, the SR of DETL is high and its NFE issignificantly less compared to those for both DE and MDE forexample 2. This clearly shows the efficiency obtained usingthe tabu list in DE (i.e., DETL).

For example 3, the SR of DE, MDE, and DETL using theRG approach are better compared to those of the FB approach.This is because most of the failed runs converged to a localsolution atx ) {0.2,-1} andy ) 0 with f ) 1.25, where twovariables (x2 and y) are at their bounds, resulting in poorperformance of the FB approach. Note that only the binaryvariable is at a bound (i.e.,y ) 1) at the global optimum. TheSR of DETL is comparable to those of DE and MDE with theRG approach and is better than those of both DE and MDEusing the FB approach for this example. The latter could bebecause of the tabu list in DETL, which avoided many functionevaluations at the local solution, thus increasing the probabilityof locating the global solution (SR are 75%, 34%, and 39% forDETL, MDE, and DE, respectively). On the other hand, NFEfor DETL is fewer than those for both DE and MDE using bothapproaches (RG and FB) for example 3.

For example 4, the SR with FB approach is found to be bettercompared to RG for all three methods. For example 5, the SRof both DE and DETL using the RG approach are bettercompared to the FB approach, whereas the SR of MDE is betterwith the FB approach compared to the RG approach. Besidesmany local minima reported by Ryoo and Sahinidis10 forexample 5, a new local minimum atx ) {0.2, 0.8, 1.5} andy) {0, 1, 1, 1} with f ) 5.636 852 is also observed in this study.The NFE for DE and MDE are the same using both approaches(RG and FB) for this example. This is because the terminationcriterion, Genmax, is satisfied before the Scmax criterion. On theother hand, DETL has high SR compared to MDE andcomparable SR to that of DE, and its NFE is less than thosefor both DE and MDE for example 5. For example 6, the SRof DE, MDE, and DETL are 100% using both RG and FBapproaches. This could be due to the presence of many globalminima, and locating one of them is sufficient to achieve the

Table 5. CPU Time (in Seconds) for NLP Examples

example number DE MDE DETL

1 0.040 0.002 0.0052 0.061 0.049 0.1013 0.166 0.109 0.2784 0.047 0.040 0.0975 0.043 0.034 0.0596 0.027 0.040 0.0477 0.066 0.049 0.1168 0.006 0.007 0.0129 0.013 0.011 0.014

10 0.043 0.037 0.05911 0.011 0.007 0.00912 0.030 0.019 0.03113 0.065 0.003 0.00314 0.069 0.026 0.02715 0.011 0.006 0.00816 0.048 0.020 0.025


best solution. The NFE of all the methods is reduced using theFB approach compared to the RG approach for this examplebecause three variables are at their bounds at the global solution,which makes the FB approach converge faster compared to RG.

For example 7, the SR is zero with the RG approach, and itis 60-75% with the FB approach for all the methods. Forexample 8, the SR is zero for both RG and FB approaches withall the methods. These results show that examples 7 and 8 arecomplex because of the associated large number of variables,constraints, and local minima14 compared to examples 1-6. Inorder to improve the performance of the algorithms, theirparameters are retuned for examples 7 and 8, and the goodparameters values obtained are as follows:A ) 0.9, CR) 0.6,NP ) 100, Genmax ) 300N, Scmax ) 20N, TR ) N × 10-2,and TLS) 100. The results with the retuned parameter valuesfor examples 7 and 8 are also given in Table 6. The SR withthe FB approach is improved and is close to 100%, whereas itis still zero with the RG approach for all the methods for thesetwo examples. This is because many decision variables (5 forboth examples 7 and 8) are at their bounds at the global solution,and handling boundary violations by the FB approach (i.e.,setting to the nearest boundary value) resulted in betterperformance. In addition to the high SR, the NFE of DETL isless compared to those for both DE and MDE for examples 7and 8.

These results show that the FB approach is comparable to orbetter than the RG approach, except for example 3. In additionto RG and FB approaches, a mixed approach is also tested forall MINLPs in this study, and the results are given in Table 6.With the mixed approach, as expected, the SR and NFE of DE,MDE, and DETL are generally between those obtained usingthe RG and FB approaches.

In general, the SR of DETL is comparable to those of DEand MDE, and its NFE is less than those of both DE and MDE.The NFE taken by local optimization is around 0.5%, 0.2%,and 0.6% of the total NFE, respectively, for DE, MDE, and

DETL. The NFE for MDE is comparable to that for DE formost of the MINLPs with an average reduction of 3% comparedto DE (Table 7). The percentage reduction in NFE for DETLcompared to DE and MDE using RG, FB, and mixed approachesis given in Table 7. On average, DETL requires 41% and 38%fewer NFE compared to the original DE and the recent MDE,respectively. The results obtained for DE and MDE implementedin this study (Table 6) are different from those reported byAngira and Babu.6 This could be due to several reasons statedunder the NLPs section along with two additional reasons: (i)

Table 6. SR and NFE of DE, MDE, and DETL Using RG, FB, and Mixed Approaches for MINLP Examples

DE MDE DETL

example number approach SR NFEa SR NFEa SR NFEa

1 RG 100 1 427+ 8 95 1 523+ 8 100 652+ 8FB 97 1 386+ 8 90 1 406+ 8 100 627+ 8mixed 97 1 459+ 8 94 1 546+ 8 100 657+ 8

2 RG 65 1 428+ 3 55 1 413+ 3 74 767+ 5FB 92 1 549+ 2 83 1 530+ 3 94 489+ 6mixed 80 1 442+ 3 78 1 484+ 3 85 710+ 5

3 RG 94 2 352+ 6 84 2 379+ 6 86 1350+ 9FB 39 1 882+ 8 34 1 819+ 8 75 1 166+ 10mixed 62 2 070+ 8 52 2 270+ 6 82 1 422+ 11

4 RG 59 2 340+ 7 57 2 361+ 6 51 2 136+ 12FB 96 1 931+ 3 89 1 684+ 3 91 718+ 8mixed 95 2 188+ 3 95 2 198+ 4 95 1 575+ 8

5 RG 84 5 620+ 4 67 5 620+ 4 94 2 793+ 11FB 80 5 620+ 4 72 5 620+ 4 84 1 764+ 13mixed 87 5 620+ 4 83 5 620+ 4 93 2 788+ 12

6 RG 100 4 020+ 4 100 4 020+ 4 100 4 020+ 7FB 100 1 216+ 4 100 1 209+ 4 100 1 001+ 4mixed 100 1 422+ 4 100 1 337+ 4 100 1 440+ 4

7 RG 0 0 0FB 75 8 020+ 36 67 7 961+ 37 60 6 943+ 78mixed 10 7 342+ 210 14 7 207+ 204 10 6 316+ 216

7b RG 0 0 0FB 100 300 100+ 8 100 296 901+ 8 96 298 721+ 17mixed 100 251 576+ 24 92 122 843+ 103 88 119 294+ 119

8b RG 0 0 0FB 99 380 097+ 633 93 401 962+ 626 96 296 989+ 646mixed 59 360 759+ 695 32 285 293+ 816 15 246 233+ 1 163

a The two numbers in each of these columns are NFE required by the method (DE, MDE, or DETL)+ NFE required for the local optimization.b Theseresults were obtained with the following parameter values:A ) 0.9, CR) 0.6, NP) 100, Genmax ) 300N, Scmax ) 20N, TR ) N × 10-2, and TLS) 100.

Table 7. Percentage Reduction in NFE for MINLP Examples

examplenumber approach

MDEcompared

to DE

DETLcompared

to DE

DETLcomparedto MDE

1 RG -6.69 54.01 56.89FB -1.43 54.45 55.09mixed -5.93 54.67 57.21

2 RG 1.05 46.05 45.48FB 1.16 68.09 67.71mixed -2.91 50.52 51.92

3 RG -1.15 42.37 43.02FB 3.33 37.78 35.63mixed -9.53 31.04 37.04

4 RG -0.85 8.479 9.25FB 12.77 62.46 56.97mixed -0.5 27.75 28.11

5 RG 0 50.14 50.14FB 0 68.40 68.40mixed 0 50.21 50.21

6 RG 0 -0.07 -0.075FB 0.57 17.62 17.15mixed 5.96 -1.26 -7.68

7a FB 1.07 81.58 81.38mixed 51.13 52.54 2.87

8a FB -5.74 21.82 26.07mixed 22.77 33.22 13.53

average 2.96 41.45 38.47

a On the basis of the results with the retuned parameter values for thesetwo examples.


the results in this study are based on 100 trials compared toonly 10 trials by Angira and Babu,6 and (ii) integers are handledby rounding the continuous variables to the nearest integers inthis study compared to truncating by Angira and Babu.6 Thisshows the need for thorough comparison of MDE and DE usinga practical stopping criterion, a common set of parameters, andmore trials as in this study to bring out the relative merits ofmethods for applications where the global minimum is unknowna priori.

The CPU time (in s) for all the methods (DE, MDE, andDETL) for MINLPs is given in Table 8. As expected, MDEtook slightly less CPU time compared to both DE and DETLfor the MINLPs tested. Though DETL took a lower NFE, itsCPU time is more than those of both DE and MDE because ofthe additional computational effort required for the tabu checkand generation steps (Figure 2). However, for applicationproblems where objective and constraints evaluation takesconsiderable CPU time, the additional computational effort fortabu checks and generating new members is expected to beinsignificant.

Conclusions

A method, namely, DETL, by combining the concepts of TSand DE, is presented for finding the global minimum of NLPand MINLP problems encountered in chemical engineering.DETL is tested for many NLP problems having different degreesof complexity. The results show that the SR of DETL is highand is comparable to those of DE and MDE in locating theglobal solutions with a lower NFE. The average reduction inNFE of DETL compared to both DE and MDE is∼35% forthe NLPs studied. DETL is then applied to MINLPs that involveprocess design and synthesis problems. The results show thatthe SR of DETL is comparable to those of DE and MDE, whileits NFE, on average, is∼40% less than those of both DE andMDE. Overall, the performance of DETL is significantly bettercompared to both DE and MDE in terms of NFE for findingthe global minimum of NLPs and MINLPs. In terms of CPUtime, MDE is better compared to both DE and DETL for theexamples tested whose function/constraints evaluation is notcomputationally intensive.

Supporting Information Available: Included as SupportingInformation are mathematical formulations for both NLPs andMINLPs in the manuscript. This material is available free ofcharge via the Internet at http://pubs.acs.org.

Nomenclature

A ) amplification factorCR ) crossover constantFB ) forcing to boundsg ) inequality constraintGenmax ) maximum number of generationsh ) equality constraintm1 ) number of equality constraintsm2 ) number of inequality constraintsMINLPs ) mixed-integer nonlinear programming problemsn ) number of continuous/real variablesNLPs ) nonlinear programming problemsNP ) population sizeNFE ) number of function evaluationsp ) total number of variablesR ) universal gas constantRG ) random generationScmax ) maximum number of successive generations without

improvement in the best function valueSR ) success ratet ) reaction timeT ) temperatureTR ) tabu radiusTLS ) tabu list sizeV ) mutant individualx ) vector of continuous/real variablesX ) target individualy ) vectory of binary variables

Superscripts

l ) lower boundu ) upper bound

Subscripts

gen) generation numbermax ) maximum numberkk ) index number for individuals in the population

Literature Cited

(1) Floudas, C. A.; Ciric, A. R.; Grossmann, I. E. Automatic synthesisof heat exchanger networks.AIChE J. 1986, 32, 276-290.

(2) Aggarwal, A.; Floudas, C. A. Global optimum search for the poolingproblem. Presented at the TIMS/ORSA Meeting, Washington, DC, 1988;pp 25-27.

(3) Shelton, M. R.; Grossmann, I. E. Optimal synthesis of integratedrefrigeration systems. I: Mixed-integer programming model.Comput. Chem.Eng.1986, 10, 577-600.

(4) Hillenbrand, J. B. Studies in the synthesis of energy-efficientevaporation systems. Ph.D. Thesis, Carnegie-Mellon University, Pittsburgh,PA, 1984.

(5) Adjiman, C. S.; Androulakis, I. P.; Floudas, C. A. Global optimizationof MINLP problems in process synthesis and design.Comput. Chem. Eng.1997, 21, S445-450.

(6) Angira, R.; Babu, B. V. Optimization of process synthesis and designproblems: A modified differential approach.Chem. Eng. Sci. 2006, 61,4707-4721.

(7) Danish, M.; Kumar, S.; Qamareen, A.; Kumar, S. Optimal solutionof MINLP problems using modified genetic algorithm.Chem. Prod. ProcessModel. 2006, 1, 1-43.

(8) Kocis, G. R.; Grossmann, I. E. Global optimization of non-convexmixed-integer non-linear programming (MINLP) problems in processsynthesis.Ind. Eng. Chem. Res.1988, 27, 1407-1421.

Table 8. CPU Time (in Seconds) for MINLP Examples

example number approach DE MDE DETL

1RG 0.011 0.004 0.026FB 0.011 0.004 0.026mixed 0.010 0.004 0.027

2RG 0.010 0.004 0.034FB 0.012 0.004 0.025mixed 0.010 0.004 0.030

3RG 0.016 0.005 0.060FB 0.012 0.009 0.059mixed 0.015 0.006 0.041

4RG 0.016 0.009 0.115FB 0.016 0.005 0.057mixed 0.015 0.006 0.043

5RG 0.032 0.026 1.079FB 0.032 0.026 0.986mixed 0.039 0.025 1.111

6RG 0.018 0.018 0.189FB 0.018 0.008 0.141mixed 0.019 0.019 0.046

7a FB 2.687 1.623 2.892mixed 2.289 0.716 4.393

8a FB 7.655 6.374 19.959mixed 7.498 4.701 16.374

a These results are using the retuned parameter values for these twoexamples.


(9) Floudas, C. A.; Aggarwal, A.; Ciric, A. R. Global optimum searchfor non-convex NLP and MINLP problems.Comput. Chem. Eng.1989,13, 1117-1132.

(10) Ryoo, H. S.; Sahinidis, N. V. Global optimization of non-convexNLPs and MINLPs with applications in process design.Comput. Chem.Eng.1995, 19, 551-566.

(11) Adjiman, C. S.; Floudas, C. A. Rigorous convex under-estimatorsfor general twice-differentiable problems.J. Global Optimization1996, 9,23-40.

(12) Torn, A.; Zilinskas, A.Global optimization; Springer-Verlag:Berlin, 1989.

(13) Das, H.; Cummings, P. T.; Levan, M. D. Scheduling of serial multi-product batch processes via simulated annealing.Comput. Chem. Eng.1990,14, 1351-1362.

(14) Salcedo, R. L. Solving non-convex non-linear programming andmixed-integer non-linear programming problems with adaptive randomsearch.Ind. Eng. Chem. Res.1992, 31, 262-273.

(15) Cardoso, M. F.; Salcedo, R. L.; Feyo de Azevedo, S.; Barbosa, D.A simulated annealing approach to the solution of MINLP problems,Comput. Chem. Eng.1997, 21, 1349-1364.

(16) Metropolis, N.; Rosenbluth, A.; Rosenbluth, M.; Teller, A.; Teller,E. Equation of state calculations by fast computing machines.J. Chem.Phys.1953, 21, 1087-1092.

(17) Nelder, J. A.; Mead, R. A simplex method for function minimiza-tion. Comput. J.1965, 7, 308-313.

(18) Jayaraman, V. K.; Kulkarni, B. D.; Karale, S.; Shelokar, P. Antcolony frame work for optimal design and scheduling of batch plants.Comput. Chem. Eng.2000, 24, 1901-1912.

(19) Yu, H.; Fang, H.; Yao, P.; Yuan, Y. A combined genetic algorithm/simulated annealing algorithm for large scale system energy integration.Comput. Chem. Eng.2000, 24, 2023-2035.

(20) Costa, L.; Oliveira, P. Evolutionary algorithms approach to thesolution of mixed integer non-linear programming problems.Comput. Chem.Eng.2001, 25, 257-266.

(21) Lin, B.; Miller, D. C. Tabu search algorithm for chemical processoptimization.Comput. Chem. Eng.2004, 28, 2287-2306.

(22) Lampinen J. A constraint handling approach for the differentialevolution algorithm. InProceedings of the Congress on EVolutionaryComputation,Honolulu, HI, 2002; IEEE Computer Society: Washington,DC, 2002; Vol. 2, pp 1468-1473.

(23) Storn, R.; Price, K. Differential evolutionsA simple and efficientheuristic for global optimization over continuous spaces.J. GlobalOptimization1997, 11, 341-359.

(24) Karaboga, N.; Cetinkaya, B. Performance comparison of geneticand differential evolution algorithms for digital FIR filter design.AdV. Inf.Syst.: Third Int. Conf., Lect. Notes Comput. Sci. 2004, 3261, 482-488.

(25) Srinivas, M.; Rangaiah, G. P. Differential evolution with tabu listfor global optimization and its application to phase equilibrium andparameter estimation problems.Ind. Eng. Chem. Res.2007, 46, 3410-3421.

(26) Srinivas, M.; Rangaiah, G. P. A study of differential Evolution andtabu search for phase stability problems. Presented at AIChE AnnualMeeting, Cincinnati, OH, 2005.

(27) Srinivas, M.; Rangaiah, G. P. A study of differential evolution andtabu Search for benchmark, phase equilibrium and phase stability problems,Comput. Chem. Eng.2007, 31, 760-772.

(28) Babu, B. V.; Angira, R. Modified differential evolution (MDE) foroptimization of non-linear chemical processes.Comput. Chem. Eng.2006,30, 989-1002.

(29) Price, K. V.; Storn, R. M.; Lampinen, J. A.Differential eVolution:A practical approach to global optimization; Springer: Berlin, 2005.

(30) Deb, K.Optimization for engineering design; Prentice Hall: NewDelhi, India, 2002.

(31) Deb, K. An efficient constraint handling method for geneticalgorithms.Comput. Methods Appl. Mech. Eng. 2000, 186, 311-318.

(32) Himmelblau, D. M.Applied nonlinear programming; McGraw-Hill: New York, 1972.

(33) Michalewicz, Z. Genetic algorithms, numerical optimization, andconstraints. InProceedings of the Sixth International Conference on GeneticAlgorithms, San Mateo, CA, 1995; Morgan Kaufmann Publishers: SanFrancisco, CA; pp 151-158.

(34) Umeda, T.; Ichikawa, A. A modified complex method for optimiza-tion, Ind. Eng. Chem. Process Des. DeV. 1971, 10, 229-236.

(35) Sahinidis, N. V.; Grossmann, I. E. Convergence properties ofgeneralized Benders decomposition.Comput. Chem. Eng.1991, 15, 481-491.

(36) Visweswaran, V.; Floudas, C. A. A global optimization algorithm(GOP) for certain classes of non-convex NLPs. II. Application of theoryand test problems.Comput. Chem. Eng.1990, 14, 1419-1434.

(37) Kocis, G. R.; Grossmann, I. E. Relaxation strategy for the structuraloptimization of process flow sheets.Ind. Eng. Chem. Res.1987, 26, 1869-1880.

(38) Floudas, C. A.Nonlinear and mixed-integer optimizationsFundamentals and applications; Oxford University Press: Oxford, U.K.,1995.

(39) Kocis, G. R.; Grossmann, I. E. A modeling and decompositionstrategy for MINLP optimization of process flow sheets.Comput. Chem.Eng.1989, 13, 797-819.

(40) Floudas, C. A.; Aggarwal, A.; Ciric, A. R. Global optimum searchfor non-convex NLP and MINLP problems.Comput. Chem. Eng.1989,13, 1117-1132.

(41) Choi, S. H.; Ko, J. W.; Manousiouthakis, V. A stochastic approachto global optimization of chemical process.Comput. Chem. Eng.1999, 23,1351-1356.

(42) Ferreira, E. C.; Lima, R.; Salcedo, R. Spreadsheets in chemicalengineering educationsA tool in process design and process integration.Int. J. Eng. Educ. 2004, 20, 928-938.

(43) Grossmann, I. E.; Sargent, R. W. H. Optimum design of multi-purpose chemical plants.Ind. Eng. Chem. Process Des. DeV. 1979, 18, 343.

ReceiVed for reView January 3, 2007ReVised manuscript receiVed July 30, 2007

AcceptedAugust 8, 2007

IE070007Q


differential evolution with tabu list for solving nonlinear and mixed-integer nonlinear programming...

Documents