Niching Diversity Estimation for Multi-modal Multi-objective Optimization
NNiching Diversity Estimation for Multi-modalMulti-objective Optimization
Yiming Peng, Hisao Ishibuchi (cid:63)
Guangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation,Department of Computer Science and Engineering,Southern University of Science and Technology, Shenzhen 518055, China.
Abstract.
Niching is an important and widely used technique in evo-lutionary multi-objective optimization. Its applications mainly focus onmaintaining diversity and avoiding early convergence to local optimum.Recently, a special class of multi-objective optimization problems, namely,multi-modal multi-objective optimization problems (MMOPs), started toreceive increasing attention. In MMOPs, a solution in the objective spacemay have multiple inverse images in the decision space, which are termedas equivalent solutions. Since equivalent solutions are overlapping (i.e.,occupying the same position) in the objective space, standard diversityestimators such as crowding distance are likely to select one of themand discard the others, which may cause diversity loss in the decisionspace. In this study, a general niching mechanism is proposed to makestandard diversity estimators more efficient when handling MMOPs. Inour experiments, we integrate our proposed niching diversity estimationmethod into SPEA2 and NSGA-II and evaluate their performance onseveral MMOPs. Experimental results show that the proposed nichingmechanism notably enhances the performance of SPEA2 and NSGA-IIon various MMOPs.
Keywords:
Niching · Diversity Estimation · Multi-modal Multi-objectiveOptimization. M ulti-objective optimization problems (MOPs), which require an optimizerto optimize multiple conflicting objective functions simultaneously, havebeen actively studied in the past few decades. For consistency, in this paper, allobjective functions are assumed to be converted to minimization problems. Dueto the trade-off between conflicting objective functions, most MOPs have a setof Pareto optimal solutions (i.e., Pareto set), which cannot be dominated by anysolutions. The projection of the Pareto set in the objective space is called thePareto front. Generally, multi-objective optimization algorithms (MOEAs) tryto find a solution set with good convergence (i.e., close to the Pareto front) and (cid:63) Corresponding author: Hisao Ishibuchi, [email protected]. a r X i v : . [ c s . N E ] J a n Yiming Peng, Hisao Ishibuchi good diversity (i.e., well-distributed over the Pareto front). Therefore, diversitymaintenance is a critical research topic across the field of evolutionary multi-objective optimization.Most MOEAs are equipped with some diversity maintenance mechanisms.Naturally, diversity estimation, which is a procedure of assigning a numericalvalue to each solution reflecting its diversity, is a prerequisite for diversity main-tenance. In the past few decades, various of diversity estimation methods havebeen developed along with the development of MOEAs. For instance, the well-known Pareto-based MOEA called NSGA-II [2] selects solutions based on theirPareto ranks (as a primary criterion) obtained from the non-dominated sort-ing procedure and their crowding distance values (as a secondary criterion). InNSGA-II, crowding distance is a diversity estimator to estimate the diversity ofeach solution for environmental selection. Another Pareto-based algorithm calledSPEA2 [19] uses the Euclidean distance from each solution to its k -th nearestneighbor in the objective space to estimate the diversity of that solution. In thegrid-based MOEA named PESA-II [1], the diversity of each solution is givenby the number of the solutions located in the same hyperbox in the objectivespace. As pointed out in [8], although the approaches used in different diver-sity estimators vary, the main idea is to estimate the diversity for a solution bymeasuring the similarity degree between that solution and other solutions in thepopulation. In this paper, the term "diversity" always refers to the diversity inthe objective space if not specified. Notice that larger diversity values are morepreferable than smaller values.Recently, multi-modal multi-objective optimization has become an active re-search topic. In MMOPs, the mapping from the decision space to the objectivespace is a many-to-one mapping instead of one-to-one mappings in standardMOPs. That is, multiple solutions with different decision values can have thesame objective values. Such kind of solutions are termed as equivalent solutions.Fig. 1 gives an example of MMOP where solutions marked with the same numberhave the same objective values. Since it is unlikely for a real-world multi-objectiveoptimization problem to have multiple solutions with exactly the same objectivevalues, the definition of equivalent solutions can be relaxed as follows [16]: Definition 1 (Equivalent Solutions).
Solutions x and x are equivalent so-lutions iff d ( F ( x ) , F ( x )) ≤ δ , where F is a vector containing all objective functions, d denotes the Euclideandistance function, and δ is a positive threshold parameter specified by the enduser.Solving MMOPs is meaningful since they are common in many real-world ap-plications, e.g., rocket engine design problems [3] and the space mission designproblems [13] both can be formulated as MMOPs. In some engineering prob-lems, obtained solutions can become infeasible or difficult to implement due todynamically changing environments and constraints [9]. In this regard, equiva-lent solutions will be able to provide alternative implementations for the decisionmaker. For this reason, when solving MMOPs, it is a good strategy to try tosearch for as many equivalent solutions as possible if no user preference is given. iching Diversity Estimation for Multi-modal Multi-objective Optimization 3 x x f f (a) Decision space. (b) Objective space. Fig. 1: Illustration of MMOP. The left and right figures show the decision andobjective spaces, respectively, and the dash lines in (a) and (b) denote the Paretoset and Pareto front, respectively. Circles marked with the same number aresolutions that have the same objective values.As pointed out in the literature [15], standard MOEAs are usually unable topreserve multiple equivalent solutions. Since equivalent solutions are located inthe same (or almost the same) position(s) in the objective space, diversity esti-mators tend to assign high density values to all of them. As a result, equivalentsolutions are usually not preferable and likely to be removed in the environmen-tal selection procedure, which leads to the failure of solving MMOPs. To tacklethis problem, we propose the use of a simple niching strategy to make stan-dard diversity estimators more efficient when handling MMOPs. Our approachis a simple, efficient, and parameterless mechanism which can be integrated intogeneral diversity estimators in existing MOEAs.The rest of the paper is organized as follows. Section 2 revisits some repre-sentative diversity estimators in MOEAs and discusses their difficulties in thehandling of MMOPs. In addition, Section 2 also introduces some existing ap-proaches for multi-modal multi-objective optimization. Next, Section 3 outlinesour proposed niching diversity estimation method. Section 4 reports experimen-tal results. Lastly, concluding remarks and suggested future research directionsare presented in Section 5.
In this section, we introduce two representative diversity estimators and discussthe difficulties they meet when handling MMOPs. Subsequently, some existingmulti-modal multi-objective optimization algorithms are reviewed.
In SPEA2 [19], each solution is assigned a density valuewhich is used to calculate its fitness value. Eq. (1) gives the density of a solution x . Density ( x ) = 1 σ k ( x ) + 2 , (1) Yiming Peng, Hisao Ishibuchi where σ k ( x ) is the distance from x to its k -th nearest neighbor in the objectivespace. In SPEA2, k is set to the square root of the total number of solutions inthe current population as a general parameter setting.Notice that in SPEA2, higher density means worse diversity in the objectivespace. Crowding distance
Crowding distance is proposed along with the NSGA-IIalgorithm [2] to preserve the diversity of the population in the objective space.The crowding distance of a solution x is given by the average side length of thehypercube constructed by its left and right neighbors in each objective. Moreprecisely, for each objective, the left and right neighbors of x are the solutionsat the left and right positions of x for that objective (i.e., in the list obtainedby sorting the population in an increasing order of the objective values of thatobjective). The crowding distance of all boundary solutions (i.e., best solutions inany objectives) are set to ∞ to ensure that they are always selected. In NSGA-II, larger crowding distance values indicate better diversity. Formally, Eq. (2)calculates the crowding distance for a solution x . Crowding-Distance ( x ) = (cid:40) ∞ , x is a boundary solution M (cid:80) Mm =1 [ f m ( x rm ) − f m ( x lm )] , otherwise , (2)where M refers to the number of objectives, and x lm and x rm are the left andright neighbors of solution x regarding the m -th objective, respectively. In most diversity estimators in MOEAs, the solution distribution in the decisionspace is out of consideration, which makes them inefficient on MMOPs. As wehave discussed in Section 1, in MMOPs, equivalent solutions have the same oralmost the same objective values. Consequently, they are usually not preferablein terms of diversity (in the objective space). For this reason, diversity estimatorsused in MOEAs are often responsible for the loss of equivalent solutions whentackling MMOPs. Fig. 2 gives an example when a diversity estimator such ascrowding distance produces undesirable effects. In Fig. 2, A and B are twoPareto optimal solutions on different (but equivalent) Pareto subsets (i.e., theupper and lower dash lines in (a)). Although A and B have similar objectivevalues, the decision maker may want to keep both of them since they representdifferent implementations (i.e., they are different in the decision space). However,a diversity estimator tends to assign bad diversity values to them due to thesmall difference between their objective values. As a result, some of them arelikely to be removed. From this example, we can see that solutions in differentregions in the decision space should be considered separately when estimatingsolution diversity for MMOPs. Following this idea, we propose a niching diversityestimation method in Section 3. iching Diversity Estimation for Multi-modal Multi-objective Optimization 5 x x A f f (a) Decision space. (b) Objective space. B BA
Fig. 2: Explanation of the diversity loss in the decision space caused by diversityestimators when handling an MMOP. The dash lines in (a) and (b) denote thePareto set and Pareto front, respectively.
In most state-of-the-art multi-modal multi-objective evolutionary algorithms(MMEAs), the diversity in the decision space is maintained by niching strate-gies. Some MMEAs extend existing niching strategies in MOEAs to enable themto maintain the diversity in the objective space as well as in the decision space.For example, in [6], Deb and Tiwari proposed one of the first MMEA calledOmni-optimizer which modifies the crowding distance to measure the diversityin the decision space and the objective space simultaneously. Yue et. al. proposeda particle swarm optimizer named MO_Ring_PSO_SCD [17] which adopts asimilar modified crowding distance and a ring topology to create a niche struc-ture. The DNEA algorithm [10] applies the fitness sharing [4] to both decisionand objective spaces and combines them into a single sharing function. SomeMMEAs are proposed with dedicated niching strategies in the decision space.Tanabe et. al. proposed a decomposition-based MMEA called MOEA/D-AD [15]where multiple solutions can be assigned to a weight vector, and a newly gener-ated solution only competes with other solutions which are assigned to the sameweight vector and neighboring to that solution in the decision space. In ourprevious study [11], we proposed another decomposition-based MMEA whichutilizes a clearing strategy in the decision space. Some MMEAs such as the algo-rithms proposed in [7] and [9] use clustering approaches to maintain the nichingstructure in the decision space.
In this section, we outline our proposed niching diversity estimation methodfor multi-modal multi-objective optimization. Here we first introduce a generalrepresentation for most diversity estimators used in MOEAs before diving intothe details of the proposed method.
Yiming Peng, Hisao Ishibuchi
Generally, for a solution x i in a solution set S , its diversity regarding S canbe expressed as follows: Diversity ( x i , S ) = Ω x j ∈ S ,j (cid:54) = i C ( x i , x j ) , (3)where C is a function which calculates the diversity contribution from a pair ofsolutions x i and x j regarding their objective values, and Ω is an aggregationfunction (e.g., sum or mean) to combine the diversity contribution from eachpair. The idea of our proposed method is straightforward: to restrict the diversityestimation within a niche. We make the following simple modifications to Eq.(3): Niching-Diversity ( x i , S ) = Diversity ( x i , S (cid:48) ) , (4)where S (cid:48) contains all solutions in S which are in the same niche as x i . In ourpaper, the closest k solutions in S to x i in the decision space are consideredas a niche.From Eq. (4), we can see that the diversity estimation for each solution islimited to its neighbors in the decision space. With the niching strategy, solutiondistribution in the decision space is taken into consideration. Take Fig. 2 as anexample, if k = 2 and crowding distance is used, the two nearest neighborsare selected for each solution (e.g., solution A ) in the decision space, and thecrowding distance is calculated using the selected neighbors in the objectivespace. Solution B is unlikely to be chosen as a neighbor of solution A (i.e., B isnot likely to be used for the crowding distance calculation of A ). In this manner,the diversity of A and B can be estimated in a desirable manner for maintainingthe decision space diversity. From this example, we can see that the proposedniching strategy can help diversity estimators in MOEAs to handle MMOPsproperly with an appropriate value of k .Compared to existing approaches we have discussed in Section 2.3, our pro-posed method does not rely on the actual implementations of diversity estima-tors. It is a general niching strategy that can be conveniently integrated intomost diversity estimators in MOEAs. In this section, we select two classical MOEAs: SPEA2 and NSGA-II to demon-strate the procedure of our proposed niching diversity estimation method intoMOEAs. The resulting algorithms are termed Niching-SPEA2 and Niching-NSGA-II, respectively.In SPEA2 and NSGA-II, diversity estimation is only involved in the envi-ronmental selection procedure although the estimated diversity values may beused in other procedures. Therefore, we only describe the modified versions ofenvironmental selection.In the environmental selection procedure of Niching-SPEA2, the nichingstrategy is applied to both fitness calculation and archive truncation as outlinedin lines 4 and 13 in Algorithm 1. In these two procedures, distance calculation iching Diversity Estimation for Multi-modal Multi-objective Optimization 7 is restricted by the niching strategy. For Niching-NSGA-II, in each generation,non-dominated sorting is employed to rank the whole population into severalfronts. Afterward, the crowding distance is computed in each front with theproposed niching strategy.
Algorithm 1:
Environmental Selection Procedure of Niching-SPEA2. input : P : input population; N : the number of survivors; output : Q : population for the next generation; /* Fitness assignment */ S ← the strength value for each solution in P ; R ← the raw fitness value for each solution in P ; D ← the niching density value for each solution in P ; for i = 1 , , . . . , | P | do F ( i ) = R ( i ) + D ( i ) ; // fitness value end /* Archive truncation */ Q ← non-dominated solutions in P ; if | Q | < N then Q ← best N solutions in P regarding their fitness values. else while | Q | > N do Repeatedly remove the solution with shortest distance to othersolutions in the same niche from Q . end end In our experiments, the performance of Niching-SPEA2 and Niching-NSGA-IIas well as the corresponding original algorithms is evaluated on ten MMOPs.Specifically, we use the SYM-PART [12], the Omni-test [6], and the MMF1–8 [17] test problems for benchmarking. For the Omni-test problem, we set thenumber of decision variables to . For the rest of test problems, the defaultparameter settings in the corresponding papers [12,17] are used. Each algorithmis evaluated on each test problem 31 times independently with population size100 and 50,000 function evaluations. The niching parameter k in Eq. (4) is setto (cid:98)√ N (cid:99) , where N is the size of S . This setting of k is based on the suggestionsin [14] for statistics and data analysis.We choose two widely used indicators: IGD + [5] and IGDX [18] to evaluatethe performance of an algorithm in the objective space and the decision space, Yiming Peng, Hisao Ishibuchi respectively. Smaller
IGD + and IGDX values indicate better proximity of theobtained solution set to the Pareto front and the Pareto set, respectively. To demonstrate the efficacy of our proposed niching strategy, first we visuallyexamine the distribution of the solution sets found by Niching-NSGA-II andits original version on the SYM-PART test problem. Non-dominated solutionsobtained from a single run of each algorithm are shown in Fig. 3 and Fig. 4.In each figure, we select the run with the median
IGD + value among all 31independent runs as a representative for visual examination. (a) Decision space. (b) Objective space. Fig. 3: Non-dominated solutions obtained by NSGA-II on SYM-PART. The blacklines show Pareto optimal solutions, and red circles show the obtained solutions.Fig. 3 (a) clearly shows that NSGA-II is poorly performed on the SYM-PARTtest problem. Most obtained solutions are distributed in the upper three Paretosubsets, while almost no solution lies on the other six Pareto subsets. This isbecause NSGA-II is a standard MOEA without diversity maintenance mecha-nisms in the decision space. In comparison, Niching-NSGA-II clearly outperformsNSGA-II as shown in Fig. 4 (a), where all nine Pareto subsets are covered. Thisexperimental result verifies that our proposed niching strategy can efficientlyprevent the loss of equivalent solutions and preserve the diversity in the decisionspace. Regarding to the distribution in the objective space, in Figs. 3 (b) and 4(b), Niching-NSGA-II slightly underperforms NSGA-II on the SYM-PART testproblem. As reported in [16], this is because equivalent solutions have small (oreven zero) contribution to the diversity in the objective space. These observa-tions suggest that there is a clear trade-off between the diversity on the Paretoset in the decision space and the diversity on the Pareto front in the objectivespace when solving MMOPs. iching Diversity Estimation for Multi-modal Multi-objective Optimization 9(a) Decision space. (b) Objective space.
Fig. 4: Obtained non-dominated solutions by Niching-NSGA-II on SYM-PART.The black lines show Pareto optimal solutions, and red circles show the obtainedsolutions.
Table 1: Statistical comparison results regarding the IGDX indicator.Mean and standard deviation of IGDX values are shown. Better resultsare highlighted.
NSGA-II Niching-NSGA-II SPEA2 Niching-SPEA2Omni-test 1.2610 ± ± + ± ± ≈ SYM-PART 7.1278 ± ± + ± ± + MMF1 0.1048 ± ± + ± ± + MMF2 0.0565 ± ± + ± ± + MMF3 0.0413 ± ± + ± ± + MMF4 0.1659 ± ± + ± ± + MMF5 0.2003 ± ± + ± ± + MMF6 0.2468 ± ± + ± ± + MMF7 0.0680 ± ± + ± ± + MMF8 1.6918 ± ± + ± ± ++ / − / ≈ baseline 10/0/0 baseline 9/0/10 Yiming Peng, Hisao Ishibuchi Table 2: Statistical comparison results regarding the
IGD + indicator.Mean and standard deviation of IGD + values are shown. Better resultsare highlighted. NSGA-II Niching-NSGA-II SPEA2 Niching-SPEA2Omni-test 0.0100 ± ± − ± ± − SYM-PART 0.0081 ± ± − ± ± − MMF1 0.0033 ± ± − ± ± − MMF2 0.0034 ± ± − ± ± − MMF3 0.0032 ± ± − ± ± − MMF4 0.0031 ± ± − ± ± − MMF5 0.0033 ± ± − ± ± − MMF6 0.0033 ± ± − ± ± − MMF7 0.0034 ± ± − ± ± − MMF8 0.0026 ± ± − ± ± − + / − / ≈ baseline 0/10/0 baseline 0/10/0 Table 1 and Table 2 present the statistical comparison results regarding theIGDX and
IGD + indicators, respectively. In each table, the Wilcoxon rank-sumtest is performed with p = 0 . to compare the performance of Niching-SPEA2and Niching-NSGA-II with their original algorithms. The symbols " + ", " − ",and " ≈ " in each table indicated that the corresponding algorithm is outperform,underperform, and tied with the baseline in the statistical comparison.Table 1 clearly shows the superiority of Niching-SPEA2 and Niching-NSGA-II in comparison to the original versions regarding the IGDX indicator. Thatis, the two modified algorithms have significantly smaller IGDX values than thecorresponding original algorithms for almost all test problems. The statisticalcomparison results further verify that the proposed niching approach can signif-icantly improve the performance of SPEA2 and NSGA-II on various MMOPs.In Table 2, we can see that the modified algorithms with the niching strategyhave worse IGD + values on all test problems than the original ones. This isconsistent with the our previous observations on Fig. 3 and Fig. 4 (i.e., thereexists a trade-off between the diversity in the decision and the objective spaces).However, from careful examinations of Table 1 and Table 2, we can see for manytest problems that large improvement of the IGDX values in Table 1 is obtainedat the cost of small deterioration of the IGD + values in Table 2. The observationabove demonstrates that the proposed niching strategy is a promising approachto multi-modal multi-objective optimization problem, whereas the handling ofthe trade-off remains an open question. In this paper, we proposed a niching diversity estimation method for multi-modal multi-objective optimization. First, we pointed out that standard diversityestimators in MOEAs meet some challenges when handling MMOPs. To address iching Diversity Estimation for Multi-modal Multi-objective Optimization 11 this issue, we proposed a general niching strategy which is applicable to existingMOEAs to enhance their performance on MMOPs. In our proposed nichingstrategy, only neighboring solutions in the decision space are involved in diversityestimation. In this manner, the proposed niching strategy is able to preventthe loss of equivalent Pareto optimal solutions. In our experimental studies, weincorporated the proposed niching strategy into two classical MOEAs: SPEA2and NSGA-II. Experimental results on ten MMOPs clearly showed that theperformance of the modified algorithms is notably improved compared to theoriginal algorithms. Currently, we employed a simple niching strategy basedon the k -th nearest neighbor. The value of k was also simply specified by thesquare root of the sample size without considering the dimensionality of thedecision space. The major contribution of this paper was to clearly illustratethat the incorporation of such a simple niching strategy significantly improvedthe performance of existing MOEAs on MMOPs. A future research directioncan be examining the effect of the value of k and to propose a more effectivespecification method. More experiments over various test problems using a widevariety of MOEAs can be conducted to examine the effects of our proposedniching mechanism on MOEAs. Moreover, another promising future researchissue is developing more sophisticated and efficient niching strategies. In thisresearch direction, the point may be how to handle the trade-off between thedecision space performance and the objective space performance. Acknowledgements
This work was supported by National Natural Science Foundation of China(Grant No. 61876075), Guangdong Provincial Key Laboratory Grant(No. 2020B121201001),the Program for Guangdong Introducing Innovative and Enterpreneurial Teams (GrantNo. 2017ZT07X386), Shenzhen Science and Technology Program (Grant No.KQTD2016112514355531), the Program for University Key Laboratory of Guang-dong Province (Grant No. 2017KSYS008).
References
1. Corne, D.W., Jerram, N.R., Knowles, J.D., Oates, M.J.: PESA-II: region-basedselection in evolutionary multiobjective optimization. In: Proc. of the 3rd AnnualConference on Genetic and Evolutionary Computation. pp. 283–290. San Francisco,USA (July 7–11 2001)2. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjec-tive genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation (2), 182–197 (April 2002)3. Fumiya, K., Tomohiro, Y., Takeshi, F.: A study on analysis of design variablesin pareto solutions for conceptual design optimization problem of hybrid rocketengine. In: Proc. of 2011 IEEE Congress on Evolutionary Computation. pp. 2558–2562. New Orleans, LA (June 5–8 2011)4. Goldberg, D.E., Richardson, J.: Genetic algorithms with sharing for multimodalfunction optimization. In: Proc. of the Second International Conference on GeneticAlgorithms. pp. 41–49. Cambridge, MA, USA (July 28–31 1987)2 Yiming Peng, Hisao Ishibuchi5. Ishibuchi, H., Masuda, H., Tanigaki, Y., Nojima, Y.: Modified distance calculationin generational distance and inverted generational distance. In: Proc. of Evolu-tionary Multi-Criterion Optimization. pp. 110–125. Guimarães, Portugal (March29–April 1 2015)6. K. Deb, S.T.: Omni-optimizer: A generic evolutionary algorithm for single andmulti-objective optimization. European Journal of Operational Research (3),1062–1087 (March 2008)7. Kramer, O., Danielsiek, H.: DBSCAN-based multi-objective niching to approxi-mate equivalent pareto-subsets. In: Proc. of the 12th Annual Conference on Ge-netic and Evolutionary Computation. pp. 503–510. Portland, Oregon, USA (July7–11 2010)8. Li, M., Yang, S., Liu, X.: Shift-based density estimation for Pareto-based algo-rithms in many-objective optimization. IEEE Transaction on Evolutionary Com-putation (3), 348–365 (June 2014)9. Lin, Q., Lin, W., Zhu, Z., Gong, M., Li, J., Coello, C.A.C.: Multimodal multi-objective evolutionary optimization with dual clustering in decision and objectivespaces. IEEE Transactions on Evolutionary Computation (2020), Early Access10. Liu, Y., Ishibuchi, H., Nojima, Y., Masuyama, N., Shang, K.: A double-nichedevolutionary algorithm and its behavior on polygon-based problems. In: Proc. ofParallel Problem Solving from Nature - PPSN XV. pp. 262–273. Coimbra, Portugal(September 8–12 2018)11. Peng, Y., Ishibuchi, H.: A decomposition-based multi-modal multi-objective opti-mization algorithm. In: Proc. of the 2020 IEEE Congress on Evolutionary Compu-tation. pp. 1–8. Glasgow, UK (July 19–24 2020)12. Rudolph, G., Naujoks, B., Preuss, M.: Capabilities of EMOA to detect and preserveequivalent Pareto subsets. In: Proc. of Evolutionary Multi-Criterion Optimization.pp. 36–50. Matsushima, Japan (March 5–8 2007)13. Schütze, O., Vasile, M., Coello Coello, C.A.: Computing the set of epsilon-efficientsolutions in multiobjective space mission design. Journal of Aerospace Computing,Information, and Communication (3), 53–70 (March 2011)14. Silverman, B.W.: Density estimation for statistics and data analysis. CRC press,London, UK (1986)15. Tanabe, R., Ishibuchi, H.: A decomposition-based evolutionary algorithm for multi-modal multi-objective optimization. In: Proc. of Parallel Problem Solving fromNature - PPSN XV. pp. 249–261. Coimbra, Portugal (September 8–12 2018)16. Tanabe, R., Ishibuchi, H.: A review of evolutionary multimodal multiobjectiveoptimization. IEEE Transactions on Evolutionary Computation (1), 193–200(February 2020)17. Yue, C., Qu, B., Liang, J.: A multiobjective particle swarm optimizer using ringtopology for solving multimodal multiobjective problems. IEEE Transactions onEvolutionary Computation (5), 805–817 (October 2018)18. Zhou, A., Zhang, Q., Jin, Y.: Approximating the set of Pareto-optimal solutionsin both the decision and objective spaces by an estimation of distribution algo-rithm. IEEE Transactions on Evolutionary Computation (5), 1167–1189 (Octo-ber 2009)19. Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength Pareto evo-lutionary algorithm. TIK-report103