Comparison of Archiving Methods in Multi-ObjectiveParticle Swarm Optimization (MOPSO): Empirical Study

Nikhil Padhye∗

Department of Mechanical EngineeringIndian Institute of Technology Kanpur

Kanpur-208016, Indianpdhye@gmail.com

ABSTRACTOver past few years, several successful proposals for handlingmulti-objective optimization tasks using particle swarm op-timization (PSO) have been made, such methods are popu-larly known as Multi-objective Particle Swarm Optimization(MOPSO). Many of these methods have focused on improv-ing characteristics like convergence, diversity and compu-tational times by proposing effective ’archiving’ and ’guideselection’ techniques. What has still been lacking is an em-pirical study of these proposals in a common frame-work. Inthis paper, an attempt to analyze these methods has beenmade; discussing their strengths and weaknesses. Combinedeffect of ’guide selection’ and ’archiving’ is also understood,and it turns out that there exist certain combinations whichperform better in terms of convergence, or diversity, or com-putational times. Finally a new hybrid proposal, by cou-pling ε-dominance with Sequential Quadratic Programming(SQP) search, has been made to achieve faster and accurateconvergence.

Categories and Subject Descriptors: I.2.8 [ArtificialIntelligence]: Problem Solving, Control Methods and Search

General Terms:Algorithms

Keywords: Multi-Objective Optimization, Archiving, Lo-cal Search

0.1 IntroductionParticle Swarm Optimization (PSO) has been successfully

used for single objective optimization, and more recentlywidely extended for multi-objective optimization. Over past,many researchers have focused on developing effective guideselection schemes (for both pbest and gbest) for (MOPSO)methods. A recent study done in [2] has focused on em-pirically comparing various guide selection schemes, whereauthors have highlighted the importance of guide selectionstrategies and their impact on convergence and diversity be-haviors of swarm. Another feature, which we regard equallyimportant, in any MOPSO method is the way in which non-dominated solutions are retained over the iterations. Usualpractice is to maintain an external archive, which is up-

∗is member at KanGal. The author greatly acknowledgesDr. C. K. Mohan’s comments. A detailed version of thisreport can be obtained by mailing the author.

Copyright is held by the author/owner(s).GECCO’09, July 8–12, 2009, Montréal, Québec, Canada. ACM 978-1-60558-325-9/09/07.

dated over iterations, to store the non-dominated solutions.Such an archive can be fixed size or unbounded. Represen-tatively speaking fixed size archiving in MOPSO, at leastto the best knowledge of the authors, has been effectivelyemployed in following ways: Clustering, Crowding dis-tance and Maximin, see [1, 3, 4]. Often based on somecriterion less crowded particles are retained. Also, some-times relaxed forms of dominance have been employed tolimit the archive size. In some studies unbounded archivehas been employed , while others have also made efficientproposals to store these non-dominated solutions.

An attempt in this paper has been made to study thecombined effect of archiving techniques and guide selectionmethods. A new proposal of coupling ε-dominance with localsearch technique is made to preserve diversity and promoteconvergence in a computationally effective way.

1. ARCHIVING METHODS AND GUIDE SE-LECTION

The three archiving techniques namely Clustering Crowd-ing distance and Maximin were combined with each ofRandom, Sigma, Wtd., ROUNDS, NWSum, Indica-tor, Crowd. dist. and Maximin global best selectiontechniques and tested on ZDT1, ZDT2, ZDT3, DTLZ1 andDTLZ2 test problems. More details on guide selection tech-niques can be found in [2]. Two performance indicators, ’Hy-pervolume’ and ’Spacing’ were employed to compare variouscombinations. For ZDT1, ZDT2 and ZDT3: pop. size 100,maximum iterations 200, tf=0.25, and Newest to updateparticles personal best [2] were used. For DTLZ1 pop size50 and maximum iterations 200 and for DTLZ2 pop. size 100and maximum iterations 500 were chosen. For DTLZ1 andDTLZ2 instead of turbulence operator (tf), velocity trigger,as proposed in [2], was employed. Average indicator val-ues and CPU time over 25 runs were computed along withstandard deviation and standard error for comparison.

For ZDT1, ZDT2 and ZDT3 problems, in terms of hyper-volume, almost all selection methods performed best withClustering. Maximin showed better hypervolume values com-pared to Crowding distance in many cases. Clustering wasalso found to achieve best spacing value. Maximin showedbetter spacing values compared to Crowding distance inmany cases. On inspection is it found that, particularly,for ZDT2 (with non-convex Pareto-front), Maximin oftenperformed poorly. This can be explained based on Maximinmethod’s property- favoring middle of convex front whichcontributes less to hypervolume value, and due to crowding

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of solutions in middle a poor spacing value results. In termsof CPU time Crowding distance was almost always found totake maximum time. Between Clustering and Maximin thelatter often took lesser time. Original Maximin approachdid not explicitly check for non-domination while updatingarchive with new population members, rather maximin val-ues for combined archive and population members were eval-uated and then the joint members with most negative valueswere retained. In current implementation first the archiveis made domination free and then Maximin strategy is em-ployed to bound the archive size. Hence one could arguethat computationally speaking Maximin is fastest.

For DTLZ1 and DTLZ2 it was found that in terms of hy-pervolume almost all the guide selection methods performedbest with Crowding distance and often worst with Maximin.On the other hand in terms of spacing, Crowding distancewas always the worst performer. The CPU times were takenleast by Crowding distance and Clustering often took max-imum time. Above observed performances can be explainedbased on the fact that Crowding distance favors to retain thesolutions at extremes and causes better convergence in theseregions. Since end points contribute relatively more to thehypervolume, and particularly more in cases of higher objec-tives, Crowding distance seems to perform well. Clusteringapproach is based solely on absolute euclidean distance inobjective space leading to even distribution of solutions andhence achieves a better spacing value. Maximin approachalso favors to retain some parts of Pareto-front, leading touneven distribution of solutions, which leads to inferior spac-ing value. Moreover, Maximin was able to preserve diversitybut did not show very good convergence.

Finally, combinations of archiving and guide selection meth-ods which perform well could be identified, for e.g., Randomwith Maximin was better in terms of computational timesand spacing as compared to Random with Clustering. Be-fore this study it would have been difficult to predict suchbehavior.

2. HYBRID APPROACHProposal for a new hybrid approach is made by combin-

ing MOPSO with Sequential Quadratic Programming (SQP,used as a local solver) along with use the of ε-dominance.Motivation for this hybrid approach arises because ε dom-inance helps to limit the archive size (making procedurefaster) while local search improves convergence. Local searchis carried out as follows. Consider such a starting pointy (having objective vector f(y) and z=f(y)) where localsolver tries to minimize the following augmented achieve-ment scalarizing function ASF:

minx ∈ S ⊂ R

n

Mmaxi=1

fi(x)−zi

fmaxi −fmin

i+ ρ

MP

j=1

fj(x)−zj

fmaxj −fmin

j

Where z=f(y) is usually reffered to as a reference pointfor local search, and fmax

i and fmini are minimum objective

values of ’best non-dominated’ set. By minimizing ASF so-lutions are projected on the Pareto-front and convergencecan be garaunteed. Thus, on applying local search archivemembers get closer to Pareto front and act as guides forother swarm members to converge accurately. In the hybridapproach, local search was applied on global archive mem-

Table 1: Test Results for Standard ApproachProb. H.V. S.D. S.E. Time(Secs)ZDT1 1.102899 0.000173 0.000035 116.197600ZDT3 1.625485 0.001701 0.000340 19.097200Oka1 16.420124 0.041584 0.008317 3.789600Oka2 120.045004 0.699837 0.139967 3.033200

Table 2: Test Results for Hybrid Approach ε=0.001Prob. H.V. S.D. S.E. Time(Secs)ZDT1 1.103568 0.000130 0.000026 78.312000ZDT3 1.629472 0.000972 0.000194 7.740000Oka1 16.444072 0.035455 0.007091 3.898000Oka2 120.271736 0.886850 0.177370 2.954400

bers periodically, in cycles of 30 iterations. The terminationcriteria for SQP solver were: either KKT error became lessthan 10−3 or a maximum of 100 SQP iterations. The av-erage performances, over 25 iterations with pop. size 100and 500 iterations, of Standard and Hybrid MOPSO havebeen listed in Tables 1 and 2. In some instances it was alsofound that hybrid approach took lesser number of functionevaluations as compared to standard approach for desiredlevel of performance measure.

3. CONCLUSIONSIn this paper we have conducted systematic empirical study

on various ’archiving’ and ’guide selection’ in MOPSO, ana-lyzing their convergence, diversity and computational times.Experiments on test problems indicated that Clusteringwas good performer, both, in terms of convergence and di-versity. But, was computationally expensive for higher ob-jectives. Maximin approach was often computationallyfastest and reported good convergence, but its performancedepended on the shape of the Pareto-front which led to im-proper distribution. Crowding distance resulted in pooruniform distribution and only showed convergence at the ex-tremes. It was also found that certain selection methods mayperform well work in combination with particular archivingtechniques. Finally, the hybrid proposal made in this paperwas found to be computationally faster and achieved betterconvergence. Such a study will allow PSO researchers tochoose and propose appropriate ’archiving’ and ’guide se-lection’ methods which can work well in combinations for awide range of problems.

4. REFERENCES[1] Xiadong Li. Better spread and convergence: Particle swarm

multiobjective optimization using the maximin fitness. InGECCO, pages 117–128, 2004.

[2] N. Padhye, J. Branke, and S. Mostaghim. A comprehensivecomparison of mopso methods: Study of convergence anddiversity- survey of state of the art. Report under review,submitted to CEC (2009).

[3] G. T. Pulido and C. A. Coello Coello. Using ClusteringTechniques to Improve the Performance of a Particle SwarmOptimizer. In Genetic and EvolutionaryComputation–GECCO 2004. Proceedings of the Genetic andEvolutionary Computation Conference. Part I, pages225–237. Springer-Verlag, Lecture Notes in Computer ScienceVol. 3102, June 2004.

[4] C. R. Raquel and P.C. Naval. An effective use of Crowdingdistance in multiobjective particle swarm optimization. InGECOO, pages 257–264, 2005.

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