Performance Assessment of Generalized Differential Evolution 3 (GDE3) with a Given Set of Problems

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1 Perormance Assessment o Generalized Dierential Evolution (GDE) with a Given Set o Problems Saku Kukkonen, Student Member, IEEE and Jouni Lampinen Abstract This paper presents results or the CEC 007 Special Session on Perormance Assessment o Multi-Objective Optimization Algorithms where Generalized Dierential Evolution (GDE) has been used to solve a given set o test problems. The set consist o 9 problems having two, three, or ive objectives. Problems have dierent properties in the sense o separability, modality, and geometry o the Pareto-ront According to the results, a near optimal set o solutions was ound in the majority o the problems. Rotated problems given caused more diiculty than the other problems. Perormance metrics indicate that obtained approximation sets were even better than provided reerence sets or many problems. I. INTRODUCTION In this paper, a general purpose Evolutionary Algorithm (EA) called Generalized Dierential Evolution (GDE) [], [] with a diversity maintenance technique suited or manyobjective problems [] has been used to solve multi-objective problems deined or the CEC 007 Special Session on Perormance Assessment o Multi-Objective Optimization Algorithms. The problems have been deined in [], where also evaluation criteria are given. The problems have two, three, or ive objectives, and the number o decision variables varies rom to 0. Diiculty o unctions vary in means o separability, modality, and geometry o the Pareto-ront. Some o the problems pose also diiculty o loosing diversity among decision variables. This paper continues with the ollowing parts: Multiobjective optimization with constraints is briely deined in Section II. Section III describes the multi-objective optimization method used to solve the given set o problems. Section IV describes experiments and inally conclusions are given in Section V. II. MULTI-OBJECTIVE OPTIMIZATION WITH CONSTRAINTS A multi-objective optimization problem (MOOP) with constraints can be presented in the orm [, p. 7]: minimize { ( x), ( x),..., M ( x)} subject to (g ( x),g ( x),...,g K ( x)) T 0. Thus, there are M unctions to be optimized and K constraint unctions. The objective o Pareto-optimization is to ind an approximation o the Pareto-ront, i.e., to ind a set o solutions that are not dominated by any other solution. Weak dominance relation between two vectors is deined such a way that x The authors are with the Department o Inormation Technology, Lappeenranta University o Technology, P.O. Box 0, FIN-8 Lappeenranta, Finland; saku.kukkonen@lut.i. weakly dominates x, i.e., x x i i : i ( x ) i ( x ). Dominance relation between two vectors is deined such a way that x dominates x, i.e., x x i x x i : i ( x ) < i ( x ). The dominance relationship can be extended to take into consideration constraint values and objective values at the same time. A constraint-domination c is deined in this paper so that x constraint-dominates x, i.e., x c x i any o the ollowing conditions is true []: x and x are ineasible and x dominates x in constraint unction violation space. x is easible and x is not. x and x are easible and x dominates x in objective unction space. The deinition or weak constraint-domination c is analogous dominance relation changed to weak dominance in the deinition above. This constraint-domination is a special case o more general concept o having goals and priorities that is presented in [7]. III. OPTIMIZATION METHOD A. Dierential Evolution The Dierential Evolution (DE) algorithm [8], [9] was introduced by Storn and Price in 99. The design principles o DE are simplicity, eiciency, and the use o loating-point encoding instead o binary numbers. As a typical EA, DE has a random initial population that is then improved using selection, mutation, and crossover operations. Several ways exist to determine a stopping criterion or EAs but usually a predeined upper limit (G max ) or the number o generations to be computed provides an appropriate stopping condition. Other control parameters or DE are the crossover control parameter (CR), the mutation actor (F ), and the population size (NP ). In each generation G, DE goes through each D dimensional decision vector x i,g o the population and creates the corresponding trial vector u i,g as ollows [0]: r,r,r {,,...,NP}, (randomly selected, except mutually dierent and dierent rom i) j rand =loor(rand i [0, ) D)+ or(j =;j D; j = j +) { i(rand j [0, ) <CR j = j rand ) u j,i,g = x j,r,g + F (x j,r,g x j,r,g) else u j,i,g = x j,i,g } --0-0/07$.00 c 007 IEEE 9

2 This is the most common DE version, DE/rand//bin. Both CR and F remain ixed during the entire execution o the algorithm. Parameter CR [0, ], which controls the crossover operation, represents the probability that an element or the trial vector is chosen rom a linear combination o three randomly chosen vectors and not rom the old vector x i,g. The condition j = j rand ensures that at least one element o the trial vector is dierent compared to the elements o the old vector. Parameter F is a scaling actor or mutation and its value range is (0, +]. In practice, CR controls rotational invariance o the search, and its small value (e.g., 0.) is practicable with separable problems while larger values (e.g., 0.9) are or non-separable problems. Parameter F controls the speed and robustness o the search, i.e., a lower value or F increases the convergence rate but it also increases the risk o getting stuck into a local optimum. Parameters CR and NP have the similar eect on the convergence rate as F has. Ater the mutation and crossover operations, the trial vector u i,g is compared to the old vector x i,g. I the trial vector has an equal or better objective value, then it replaces the old vector in the next generation. This can be presented as ollows in the case o minimization o an objective [0]: { ui,g i ( u x i,g+ = i,g ) ( x i,g ). x i,g otherwise DE is an elitist method since the best population member is always preserved and the average objective value o the population will never deteriorate. B. Generalized Dierential Evolution The irst version o Generalized Dierential Evolution (GDE) extended DE or constrained multi-objective optimization, and it modiied only the selection rule o the basic DE []. The basic idea in the selection rule o GDE is that the trial vector is selected to replace the old vector in the next generation i it weakly constraint-dominates the old vector. There was no explicit sorting o non-dominated solutions [, pp. ] during the optimization process or any mechanism or maintaining the distribution and extent o solutions. Also, there was no extra repository or nondominated solutions. The second version, GDE, made the selection based on crowdedness when the trial and old vector were easible and non-dominating each other in the objective unction space []. This improved the extent and distribution o the obtained set o solutions but slowed down the convergence o the overall population because it avored isolated solutions ar rom the Pareto-ront until all the solutions were converged near the Pareto-ront. The third and latest version is GDE [], []. Besides the selection, another part o the basic DE has also been modiied. Now, in the case o easible and non-dominating solutions, both vectors are saved or the population o next Notation means that F is larger than 0 and upper limit is in practice around although there is no hard upper limit. generation. Beore continuing to the next generation, the size o the population is reduced using non-dominated sorting and pruning based on diversity preservation. The pruning technique used in the original GDE is based on crowding distance, which provides a good crowding estimation in the case o two objectives. However, crowding distance ails to approximate crowdedness o solutions when the number o objectives is more than two []. Since, the provided problem set in [] consists o problems with more than two objectives, a more general diversity maintenance technique proposed in [] was used. The technique is based on a crowding estimation using the nearest neighbors o solutions in Euclidean sense, and an eicient nearest neighbors search technique. All the GDE versions handle any number o M objectives and any number o K constraints, including the cases where M = 0 (constraint satisaction problem) and K = 0 (unconstrained problem). When M = and K = 0, the versions are identical to the original DE, and this is why they are reerred as Generalized DEs. IV. EXPERIMENTS A. Coniguration GDE and the given problems were implemented in the ANSI-C programming language and compiled with the GCC compiler. The hardware was an ordinary PC with. GHz CPU & GB RAM, and the operating system was Linux. In the case o boundary constraint violations, violating variable values were relected back rom the violated boundary using ollowing rule beore the selection operation o GDE: x (lo) j u j,i,g i u j,i,g <x (lo) j u j,i,g = x (up) j u j,i,g i u j,i,g >x (up), j u j,i,g otherwise where x (lo) j and x (up) j are lower and upper limits respectively or a decision variable x j. B. Parameter Setting Along stopping criterion and size o the population (NP ), GDE has two control parameters (CR and F ) as described in Section III-A, where the eect and ranges o these are also given. As a rule o thumb in single-objective optimization, i nothing is known about the problem in hand then suitable initial control parameter values are CR =0.9 and F =0.9, and NP = D...0 D, where D is the number o decision variables o the problem []. For an easy problem (e.g., moderately multimodal and low dimensional), a small value o NP is suicient but with diicult problems, a large value o NP is recommended in order to avoid stagnation to a local optimum. In general, increase o control parameter values, will also increase the number o unction evaluations (FES) needed. Dependency between NP and FES needed is linear while FES needed increases more rapidly along CR and F []. I values o F and/or NP are too low, search is prone to stagnate to a local optimum (with very low control IEEE Congress on EvolutionaryComputation (CEC 007)

3 parameter values the search converges without the selection pressure). In the case o multi-objective optimization and conlicting objectives, lower control parameter values (e.g., 0.) orcr and F can be used than in single-objective optimization because conlicting objectives already maintain diversity and restrain the search speed. This has been noted in [], [], where the eect o the control parameters has been studied empirically. Also, i the complexity o objectives dier (e.g., as in the case o the ZDT problems [, pp. 0]), then a high value or CR might lead to premature convergence with respect to one objective compared to another. The value o NP can be selected in same way as in single-objective optimization or it can be selected according to a desired approximation set size o the Pareto-optimal ront. To keep the setup as simple as possible, same set o control parameter values were used or all the problems. It would had been also possible to apply some kind o dependency on the number o objectives and/or the number o decision variables. As well, it would had been possible to use some control parameter adaptation strategy as in [7] [9]. However, these were not applied, because then it would had been unclear, how parameter adjustment vs. the optimization algorithm itsel contributes to the results. The control parameter values used were CR = 0., F =0., NP = 00, and G max = 999. First two control parameter values were obtained by a couple o preliminary tests with the problems. Value o CR was relatively low but suitable because most o the problems were separable and because the complexity o objectives dier or the problems modiied rom the ZDT problems []. Value o F is a compromise between speed and robustness. Value F =0.has been noted to be especially suitable or the ZDT problem because o equally spaced local optima [], []. The size o the population was set according to the smallest desired approximation set size given in []. With chosen NP and G max values, FES is exactly , which was an upper limit given in []. In [], approximation sets o dierent sizes were demanded or dierent problems. In GDE, the size o the approximation set is usually same as NP.Now,NP was kept ixed or all the problems, and solutions or the approximation set were collected during generations. Populations o 00 last generations (9 last generations in the case o 000 unction evaluations) were merged together and nondominated solutions were selected rom this merged set o solutions. I the size o non-dominated set was larger than the desired approximation set size, then the set was reduced to desired size using the pruning technique described in []. C. Results o Experiments The problems given in [] were solved times and achieved results are presented in Tables I VII and Figs.. Tables II VII show the best, the worst, median, mean, and standard deviation values o two perormance indicators ater dierent FES. Binary indicators used were R [0] and Hypervolume indicator (I H) []. When these are used to compare against a reerence set, then a smaller indicator value express better perormance (value 0 indicates equal perormance respect to the reerence set). According to the R indicator values in Tables II IV, a better approximation set than the given reerence set (i.e., negative indicator value) was ound or problems OKA, S ZDT, S ZDT, WFG (M=), WFG8 (M=,), and WFG9 (M=). According to the Hypervolume indicator values in Tables V VII, a better approximation set than the given reerence set was ound or problems OKA, S ZDT, S DTLZ (M=), WFG (M=), WFG8 (M=,), WFG9 (M=,), and R DTLZ (M=). Covered sets (CS) metric [] measures number o covered Pareto-subsets in the decision variable space. The SYM- PART problem has been designed to have several Paretosubsets in the the decision variable space mapping into a single Pareto-set in the objective space. CS values or the SYMPART problem are shown in Table I and these indicate that solutions converged into a single Pareto-subset in the decision variable space. This is not surprising since GDE does not explicitly maintain diversity in the decision variable space but in the objective space according to common goals o multi-objective optimization []. TABLE I THE RESULTS FOR COVERED SETS CS FOR TEST FUNCTION SYMPART FES e+ e+ e+ Best.0000e e e+00 Median.0000e e e+00 Worst e e e+00 Mean.0000e e e+00 Std 0.000e e e+00 Attainment suraces [] in the case o two and three objectives are shown in Figs. and. Results in Fig. indicate that the Pareto-optimal ront is ound reliably in all the two-objective cases except with the ZDT problem modiications and S ZDT. According to Fig., the rotated version o ZDT (R ZDT) is harder to solve than the unrotated version (S ZDT). 00 % attainment surace or S ZDT is a single point (, ), thus sometimes the result can be a single solution or this problem. Dierent control parameter values would give dierent results. Attainment suraces or the three-objective problems in Fig. indicate good perormance in the most cases. The rotated DTLZ problem (R DTLZ) seems to be the most diicult one, and also S DTLZ causes some diiculty. Figure illustrates pairwise interaction o objective unctions in the case o the ive-objective WFG8 and WFG9 problems. The results indicate good coverage o the Paretooptimal ront since the approximation sets appear to be close to the true Pareto-ront and also the area o the Pareto-ront is well covered. D. Algorithm Complexity GDE is well scalable algorithm with the simple genetic operations o DE. Thereore, also problems with a large 007 IEEE Congress on EvolutionaryComputation (CEC 007) 9

4 TABLE II THE RESULTS FOR R INDICATOR ON TEST FUNCTIONS 7 FES. OKA. SYMPART. S ZDT. S ZDT. S ZDT. R ZDT 7. S ZDT Best -9.7e-0.99e-0.9e-0 8.0e-0.97e-0 9.9e-0.0e-0 Median -.70e-0.7e-0.9e-0 9.8e-0 7.7e-0.779e-0.e-0 e+ Worst 8.70e-0.80e-0.07e-0.08e e-0.8e-0.880e-0 Mean.7e-0.8e-0.e-0 9.8e-0 7.e e-0.e-0 Std.978e-0.089e-0.8e-0.000e-0.7e-0.e-0.7e-0 Best -.00e-0.e-0.7e-0.9e-0.8e-0.07e-0.799e-0 Median -9.8e-0.e-0.00e-0.88e-0.8e-0 7.7e-0.99e-0 e+ Worst -8.00e-0.877e-0.9e-0.09e-0 9.9e-0.7e-0.e-0 Mean -9.80e-0.8e-0.e-0.8e-0.97e-0 8.e-0.990e-0 Std.0e-0 7.0e-0.e-0.0e e-0.e-0 7.8e-0 Best -.0e-0.8e-0.98e-08.8e e-09.7e e-0 Median -.09e-0.07e-0.7e e-0 -.e e e-0 e+ Worst -.00e-0.99e-0.0e-0.00e-0 7.7e-0.99e e-0 Mean -.08e-0.9e-0.8e-0.e-0.e-0.77e e-0 Std.7e-0.0e-07.09e-0.088e-0.88e-0.0e-0.87e- TABLE III THE RESULTS FOR R INDICATOR ON TEST FUNCTIONS 8 WHEN M= FES 8. S DTLZ 9. R DTLZ 0. S DTLZ. WFG. WFG8. WFG9 Best.0e-0.7e-0.0e-0.8e-0 -.e e-0 Median.90e-0.908e-0.880e e-0 -.0e e-0 e+ Worst.e-0.978e-0 7.0e-0 8.0e e-0.7e-0 Mean.098e-0.00e-0.880e e e e-0 Std.e-0.7e-0.07e-0.98e-0.8e-0.0e-0 Best.9e e-0.007e-0.07e e e-0 Median.7e-0.7e-0.79e-0.899e-0 -.9e e-0 e+ Worst 9.78e-0.708e-0.e-0.00e e-0 -.e-0 Mean.0877e-0.7e-0.79e-0.7e-0 -.9e-0-8.9e-0 Std.0e-0.097e-0.70e-0.08e-0.e-0.e-0 Best.7e-0.87e-0.88e e e e-0 Median.877e-0.0e-0.77e-07.7e-0 -.8e e-0 e+ Worst.99e-0.897e-0.e-0.99e e-0 -.8e-0 Mean.00e-0.897e-0.90e-07.77e e-0-9.e-0 Std.90e-0 7.9e-0.e-07.77e-0.e-0.979e-0 TABLE IV THE RESULTS FOR R INDICATOR ON TEST FUNCTIONS 8 WHEN M= FES 8. S DTLZ 9. R DTLZ 0. S DTLZ. WFG. WFG8. WFG9 Best.87e-0.89e-0.800e-0.0e-0.97e-0.99e-0 Median.0e-0.9e-0.090e-0.7e-0.7e-0.87e-0 e+ Worst.090e-0.0e-0.7e-0.78e-0 8.e-0.e-0 Mean.07e-0.9e-0.e-0.8e-0.878e-0 7.0e-0 Std.7e-0.08e-0.9e-0.78e-0.08e-0.99e-0 Best.9e e-0 8.9e-0.8e e-0.8e-0 Median.789e-0.8e-0.80e-0.8e e-0.7e-0 e+ Worst.78e-0 7.9e-0.e-0.88e-0 -.e-0.987e-0 Mean.9e-0.78e-0.9e-0.e-0-7.7e-0.7e-0 Std 7.9e e-0.0e-0.700e-0 8.e e-0 Best.0e-0.8e-0 8.0e-09.9e-0 -.9e-0.0e-0 Median.08e-0.8e-0.7e-07.00e-0 -.e-0.8e-0 e+ Worst.9e e-0.07e e-0 -.0e-0.987e-0 Mean.e-0.e-0.e e-0 -.e-0.998e-0 Std.0e-0.08e-0.79e-07.9e-0.99e-0 7.0e IEEE Congress on EvolutionaryComputation (CEC 007)

5 TABLE V THE RESULTS FOR HYPERVOLUME INDICATOR I H ON TEST FUNCTIONS 7 FES. OKA. SYMPART. S ZDT. S ZDT. S ZDT. R ZDT 7. S ZDT Best -8.8e-0.07e-0.e-0.090e-0.780e-0.987e-0.0e-0 Median -.07e-0 7.7e-0.e-0.e-0.8e-0.98e-0.7e-0 e+ Worst.e-0.08e e-0.79e-0.e e-0.e-0 Mean.79e e-0.e-0.7e-0.90e-0.07e-0.97e-0 Std.879e-0.7e-0.8e-0.e-0.7e-0.0e-0.8e-0 Best -.7e-0.98e-0.87e-0.97e-0.708e-0.e-0.98e-0 Median -.87e-0.7e-0.8e-0.9e-0.008e e-0.8e-0 e+ Worst e-0.e e-0.80e-0.77e-0.88e-0.88e-0 Mean -.0e e-0.79e-0.8e-0.078e-0.07e-0.900e-0 Std.7e-0.e-0 9.0e-0.e-0.9e-0.9e-0.90e-0 Best -.9e-0.90e-0.7e-0.9e-0 8.0e e-0 -.e-0 Median -.80e-0.079e-0.707e-0.0e e-07.e e-0 e+ Worst -.08e-0.90e-0.00e-0.78e-0.9e-0.09e e-0 Mean -.7e-0.77e-0.78e-0.0e-0.790e-0.90e e-0 Std.e-0.8e-07.0e-0.78e-0.70e-0.9e-0.0e-07 TABLE VI THE RESULTS FOR HYPERVOLUME INDICATOR I H ON TEST FUNCTIONS 8 WHEN M= FES 8. S DTLZ 9. R DTLZ 0. S DTLZ. WFG. WFG8. WFG9 Best.90e e-0.88e-0.e-0-8.8e e-0 Median.7e-0.090e-0 7.e-0.079e-0-7.9e-0 -.9e-0 e+ Worst.708e-0.87e-0.00e-0.98e e-0.8e-0 Mean.88e-0.08e-0 7.7e-0.00e-0-7.0e-0 -.9e-0 Std.900e-0.e-0.00e-0.e e-0.889e-0 Best.9e-0.00e-0.0e-0.87e e e-0 Median.07e-0.97e-0.e-0.0e e-0 -.9e-0 e+ Worst.0e-0 8.9e e-0.8e-0 -.e e-0 Mean.7e-0.89e-0.99e-0.89e e-0 -.e-0 Std 8.e-0.7e-0.9e-0.98e-0.008e-0.0e-0 Best -.9e-0.880e-0.9e e e e-0 Median -7.09e-0 7.e-0 7.7e e e e-0 e+ Worst.98e-0.789e-0.08e-08.07e e e-0 Mean -.88e-0 8.7e-0.98e-09.77e-0 -.7e e-0 Std.87e-0.709e-0.00e-09.77e e e-0 TABLE VII THE RESULTS FOR HYPERVOLUME INDICATOR I H ON TEST FUNCTIONS 8 WHEN M= FES 8. S DTLZ 9. R DTLZ 0. S DTLZ. WFG. WFG8. WFG9 Best 8.e-0.07e-0.87e-0.9e e-0 7.0e-0 Median.7e-0.00e-0.7e-0.e-0 -.e-0.99e-0 e+ Worst.7e-0.8e-0.9e-0.e e-0.08e-0 Mean.77e-0.97e-0.9e-0.9e-0 -.9e-0.8e-0 Std.99e-0 8.e-0.0e-0.88e-0.789e-0.79e-0 Best.08e-0-8.7e-0.98e-0.99e e-0 -.e-0 Median.0e-0.08e-0 7.8e-0.e-0 -.9e e-0 e+ Worst.0e-0 8.0e-0.788e-0.708e-0 -.8e e-0 Mean.778e-0 -.7e e-0.7e e e-0 Std.e-0.0e-0.8e-0.907e-0.99e-0 9.7e-0 Best.e e-0.880e-.997e-0 -.7e-0 -.0e-0 Median 9.9e e-0.9e-.99e-0 -.0e-0 -.0e-0 e+ Worst.08e e-0.808e e-0 -.8e e-0 Mean.000e-0 -.e-0.9e-0.9e e-0 -.e-0 Std.87e-0.7e e-0.0e-0 9.9e-0.9e IEEE Congress on EvolutionaryComputation (CEC 007) 97

6 OKA 0. 0% attainment surace 0. 0% attainment surace 00% attainment surace Pareto optimal ront SYMPART 0% attainment surace 0% attainment surace 00% attainment surace Pareto optimal ront S_ZDT 0% attainment surace 0% attainment surace 00% attainment surace Pareto optimal ront.8. S_ZDT % attainment surace 0% attainment surace. 00% attainment surace Pareto optimal ront S_ZDT 0% attainment surace 0% attainment surace 00% attainment surace Pareto optimal ront R_ZDT 0% attainment surace 0% attainment surace 00% attainment surace Pareto optimal ront S_ZDT % attainment surace 0% attainment surace 00% attainment surace Pareto optimal ront Fig.. Pareto-optimal ront and 0%, 0%, 00% attainment suraces ater e+ FES on test unctions IEEE Congress on EvolutionaryComputation (CEC 007)

7 S_DTLZ R_DTLZ S_DTLZ WFG9. WFG8 0. Fig.. 00 WFG % attainment suraces ater e+ FES on test unctions 8 (M = ) Fig Upper diagonal plots are or WFG8 (M = ) and lower diagonal plots are or WFG9 (M = ). 007 IEEE Congress on Evolutionary Computation (CEC 007) 99

8 number o decision variables and/or a large population size as in [] are solvable in reasonable time. The most complex operation in GDE ( is non-dominated ) sorting, which time complexity is O N log M N []. The time complexity o GDE was measured according to instructions given in [], and observed CPU times are reported in Table VIII. TABLE VIII COMPUTATIONAL COMPLEXITY: TIS MEAN TIME FOR EVALUATING ALL THE PROBLEMS 0000 TIMES AND T IS MEAN TIME FOR SOLVING ALL THE PROBLEMS USING GDE AND 0000 FUNCTION EVALUATIONS T T (T T ) /T. s.78 s 0.88 V. CONCLUSIONS Results o Generalized Dierential Evolution (GDE) or the CEC 007 Special Session on Perormance Assessment o Multi-Objective Optimization Algorithms have been reported. The problems given were solved with the same ixed control parameter settings, i.e., there was no parameter adaption based on problem characteristic or other criteria. According to the results, GDE perormed well especially with the problems without artiicial rotation according to the attainment suraces obtained. Worse perormance with rotated problems was probably due to selected control parameter values. In several cases, the perormance metrics indicate that the obtained approximation sets were even better than the given reerence sets. REFERENCES [] S. Kukkonen and J. Lampinen, GDE: The third evolution step o Generalized Dierential Evolution, in Proceedings o the 00 Congress on Evolutionary Computation (CEC 00), Edinburgh, Scotland, Sept 00, pp. 0. [Online]. Available: [] S. Kukkonen and K. Deb, Improved pruning o non-dominated solutions based on crowding distance or bi-objective optimization problems, in Proceedings o the 00 Congress on Evolutionary Computation (CEC 00), Vancouver, BC, Canada, July 00, pp [], A ast and eective method or pruning o non-dominated solutions in many-objective problems, in Proceedings o the 9th International Conerence on Parallel Problem Solving rom Nature (PPSN IX),, Reykjavik, Iceland, Sept 00, pp.. [] V. L. Huang, A. K. Qin, K. Deb, E. Zitzler, P. N. Suganthan, J. J. Liang, M. Preuss, and S. Huband, Problem deinitions or perormance assessment on multi-objective optimization algorithms, School o EEE, Nanyang Technological University, Singapore, 9798, Technical Report, January 007. [Online]. Available: iles/cec- 07/CEC-07-TR--Feb.pd, [] K. Miettinen, Nonlinear Multiobjective Optimization. Boston: Kluwer Academic Publishers, 998. [] J. 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