An improved epsilon constraint-handling method in MOEA/D for CMOPs with large infeasible regions

This paper proposes an improved epsilon constraint-handling mechanism and combines it with a decomposition-based multi-objective evolutionary algorithm (MOEA/D) to solve constrained multi-objective optimization problems (CMOPs). The proposed constrained multi-objective evolutionary algorithm (CMOEA)...

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Published in:	Soft computing (Berlin, Germany) Vol. 23; no. 23; pp. 12491 - 12510
Main Authors:	Fan, Zhun, Li, Wenji, Cai, Xinye, Huang, Han, Fang, Yi, You, Yugen, Mo, Jiajie, Wei, Caimin, Goodman, Erik
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2019 Springer Nature B.V
Subjects:	Artificial Intelligence Collaboration Computational Intelligence Constraints Control Decomposition Engineering Evolutionary algorithms Mathematical Logic and Foundations Mechatronics Methodologies and Application Methods Multiple objective analysis Optimization Performance evaluation Robotics Violations Constrained multi-objective optimization Robot gripper optimization Constrained multi-objective evolutionary algorithms Epsilon constraint handling
ISSN:	1432-7643, 1433-7479
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Abstract	This paper proposes an improved epsilon constraint-handling mechanism and combines it with a decomposition-based multi-objective evolutionary algorithm (MOEA/D) to solve constrained multi-objective optimization problems (CMOPs). The proposed constrained multi-objective evolutionary algorithm (CMOEA) is named MOEA/D-IEpsilon. It adjusts the epsilon level dynamically according to the ratio of feasible to total solutions in the current population. In order to evaluate the performance of MOEA/D-IEpsilon, a new set of CMOPs with two and three objectives is designed, having large infeasible regions (relative to the feasible regions), and they are called LIR-CMOPs. Then, the 14 benchmarks, including LIR-CMOP1-14, are used to test MOEA/D-IEpsilon and four other decomposition-based CMOEAs, including MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and CMOEA/D. The experimental results indicate that MOEA/D-IEpsilon is significantly better than the other four CMOEAs on all of the test instances, which shows that MOEA/D-IEpsilon is more suitable for solving CMOPs with large infeasible regions. Furthermore, a real-world problem, namely the robot gripper optimization problem, is used to test the five CMOEAs. The experimental results demonstrate that MOEA/D-IEpsilon also outperforms the other four CMOEAs on this problem.
AbstractList	This paper proposes an improved epsilon constraint-handling mechanism and combines it with a decomposition-based multi-objective evolutionary algorithm (MOEA/D) to solve constrained multi-objective optimization problems (CMOPs). The proposed constrained multi-objective evolutionary algorithm (CMOEA) is named MOEA/D-IEpsilon. It adjusts the epsilon level dynamically according to the ratio of feasible to total solutions in the current population. In order to evaluate the performance of MOEA/D-IEpsilon, a new set of CMOPs with two and three objectives is designed, having large infeasible regions (relative to the feasible regions), and they are called LIR-CMOPs. Then, the 14 benchmarks, including LIR-CMOP1-14, are used to test MOEA/D-IEpsilon and four other decomposition-based CMOEAs, including MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and CMOEA/D. The experimental results indicate that MOEA/D-IEpsilon is significantly better than the other four CMOEAs on all of the test instances, which shows that MOEA/D-IEpsilon is more suitable for solving CMOPs with large infeasible regions. Furthermore, a real-world problem, namely the robot gripper optimization problem, is used to test the five CMOEAs. The experimental results demonstrate that MOEA/D-IEpsilon also outperforms the other four CMOEAs on this problem.
Author	Mo, Jiajie Cai, Xinye Goodman, Erik Fan, Zhun Huang, Han You, Yugen Wei, Caimin Fang, Yi Li, Wenji
Author_xml	– sequence: 1 givenname: Zhun surname: Fan fullname: Fan, Zhun organization: Department of Electronic Engineering, Shantou University – sequence: 2 givenname: Wenji surname: Li fullname: Li, Wenji organization: Department of Electronic Engineering, Shantou University – sequence: 3 givenname: Xinye surname: Cai fullname: Cai, Xinye email: Xinye@nuaa.edu.cn organization: College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics – sequence: 4 givenname: Han surname: Huang fullname: Huang, Han organization: School of Software Engineering, South China University of Technology – sequence: 5 givenname: Yi surname: Fang fullname: Fang, Yi organization: Department of Electronic Engineering, Shantou University – sequence: 6 givenname: Yugen surname: You fullname: You, Yugen organization: Department of Electronic Engineering, Shantou University – sequence: 7 givenname: Jiajie surname: Mo fullname: Mo, Jiajie organization: College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics – sequence: 8 givenname: Caimin surname: Wei fullname: Wei, Caimin organization: Department of Mathematics, Shantou University – sequence: 9 givenname: Erik surname: Goodman fullname: Goodman, Erik organization: BEACON Center for the Study of Evolution in Action, Michigan State University
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Keywords	Constrained multi-objective optimization Robot gripper optimization Constrained multi-objective evolutionary algorithms Epsilon constraint handling
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References	Coello CoelloCATheoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the artComput Methods Appl Mech Eng200219111–1212451287187768410.1016/S0045-7825(01)00323-1 LiJDA two-step rejection procedure for testing multiple hypothesesJ Stat Plann Inference2008138615211527242728710.1016/j.jspi.2007.04.032 Zitzler E, Laumanns M, Thiele L (2001) SPEA2: Improving the strength Pareto evolutionary algorithm. TIK-report 103 CaiXHuZFanZA novel memetic algorithm based on invasive weed optimization and differential evolution for constrained optimizationSoft Comput201317101893191010.1007/s00500-013-1028-4 JiangSZhangJOngYSZhangANTanPSA simple and fast hypervolume indicator-based multiobjective evolutionary algorithmIEEE Trans Cybern201545102202221310.1109/TCYB.2014.2367526 Takahama T, Sakai S (2006) Constrained optimization by the ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} constrained differential evolution with gradient-based mutation and feasible elites. In: 2006 IEEE international conference on evolutionary computation. IEEE, pp 1–8 DebKMulti-objective optimization using evolutionary algorithms2001LondonWiley0970.90091 DerracJGarcíaSMolinaDHerreraFA practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithmsSwarm Evol Comput20111131810.1016/j.swevo.2011.02.002 RunarssonTPYaoXStochastic ranking for constrained evolutionary optimizationIEEE Trans Evol Comput20004328429410.1109/4235.873238 HolmSA simple sequentially rejective multiple test procedureScand J Stat1979665705385970402.62058 Corne DW, Jerram NR, Knowles JD, Oates MJ (2001) PESA-II: region-based selection in evolutionary multiobjective optimization. In: Proceedings of the 3rd annual conference on genetic and evolutionary computation. Morgan Kaufmann, pp 283–290 ZhangQLiHMOEA/D: a multiobjective evolutionary algorithm based on decompositionIEEE Trans Evol Comput200711671273110.1109/TEVC.2007.892759 RomDMA sequentially rejective test procedure based on a modified Bonferroni inequalityBiometrika1990773663665108786010.1093/biomet/77.3.663 CaiXYangZFanZZhangQDecomposition-based-sorting and angle-based-selection for evolutionary multiobjective and many-objective optimizationIEEE Trans Cybern20174792824283710.1109/TCYB.2016.2586191 Yang Z, Cai X, Fan Z (2014) Epsilon constrained method for constrained multiobjective optimization problems: some preliminary results. In: Proceedings of the Companion Publication of the 2014 Annual Conference on Genetic and Evolutionary Computation, ACM, pp 1181–1186 BeumeNNaujoksBEmmerichMSMS-EMOA: multiobjective selection based on dominated hypervolumeEur J Oper Res200718131653166910.1016/j.ejor.2006.08.008 Datta R, Deb K (2011) Multi-objective design and analysis of robot gripper configurations using an evolutionary-classical approach. In: Conference on genetic and evolutionary computation, pp 1843–1850 Zitzler E, Künzli S (2004) Indicator-based selection in multiobjective search. In: International conference on parallel problem solving from nature. Springer, pp 832–842 Fan Z, Li W, Cai X, Li H, Hu K, Zhang Q, Deb K, Goodman ED (2016) Difficulty adjustable and scalable constrained multi-objective test problem toolkit. arXiv preprint arXiv:1612.07603 BaderJZitzlerEHypE: an algorithm for fast hypervolume-based many-objective optimizationEvol Comput2011191457610.1162/EVCO_a_00009 ZitzlerEThieleLMultiobjective evolutionary algorithms: a comparative case study and the strength Pareto approachIEEE Trans Evol Comput19993425727110.1109/4235.797969 JanMAKhanumRAA study of two penalty-parameterless constraint handling techniques in the framework of MOEA/DAppl Soft Comput201313112814810.1016/j.asoc.2012.07.027 FinnerHOn a monotonicity problem in step-down multiple test proceduresJ Am Stat Assoc199388423920923124294210.1080/01621459.1993.10476358 HollandBSCopenhaverMDAn improved sequentially rejective Bonferroni test procedureBiometrics19874341742389741010.2307/2531823 Zhang Q, Zhou A, Zhao S, Suganthan PN, Liu W, Tiwari S (2008) Multiobjective optimization test instances for the CEC 2009 special session and competition. University of Essex, Colchester, UK and Nanyang Technological University, Singapore, special session on performance assessment of multi-objective optimization algorithms, Technical report 264 LiuHLGuFZhangQDecomposition of a multiobjective optimization problem into a number of simple multiobjective subproblemsIEEE Trans Evol Comput201418345045510.1109/TEVC.2013.2281533 DebKPratapAAgarwalSMeyarivanTA fast and elitist multiobjective genetic algorithm: NSGA-IIIEEE Trans Evol Comput20026218219710.1109/4235.996017 RunarssonTPYaoXSearch biases in constrained evolutionary optimizationIEEE Trans Syst Man Cybern Part C Appl Rev200535223324310.1109/TSMCC.2004.841906 DunnOJMultiple comparisons among meansJ Am Stat Assoc196156293526412495210.1080/01621459.1961.10482090 LiHZhangQMultiobjective optimization problems with complicated pareto sets, MOEA/D and NSGA-IIIEEE Trans Evol Comput200913228430210.1109/TEVC.2008.925798 CaiXLiYFanZZhangQAn external archive guided multiobjective evolutionary algorithm based on decomposition for combinatorial optimizationIEEE Trans Evol Comput201519450852310.1109/TEVC.2014.2350995 ČrepinšekMLiuSHMernikMExploration and exploitation in evolutionary algorithms: a surveyACM Comput Surv20134533510.1145/2480741.2480752 Asafuddoula M, Ray T, Sarker R, Alam K (2012) An adaptive constraint handling approach embedded MOEA/D. In: 2012 IEEE Congress on Evolutionary Computation. IEEE, pp 1–8 Mezura-MontesECoello CoelloCAConstraint-handling in nature-inspired numerical optimization: past, present and futureSwarm Evol Comput20111417319410.1016/j.swevo.2011.10.001 MiettinenKNonlinear multiobjective optimization1999BerlinSpringer0949.90082 BosmanPAThierensDThe balance between proximity and diversity in multiobjective evolutionary algorithmsIEEE Trans Evol Comput20037217418810.1109/TEVC.2003.810761 HochbergYA sharper Bonferroni procedure for multiple tests of significanceBiometrika198875480080299512610.1093/biomet/75.4.800 HommelGA stagewise rejective multiple test procedure based on a modified Bonferroni testBiometrika198875238338610.1093/biomet/75.2.383 HuZCaiXFanZAn improved memetic algorithm using ring neighborhood topology for constrained optimizationSoft Comput201318102023204110.1007/s00500-013-1183-7 HubandSHingstonPBaroneLWhileLA review of multiobjective test problems and a scalable test problem toolkitIEEE Trans Evol Comput200610547750610.1109/TEVC.2005.861417 DebKJainHAn evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraintsIEEE Trans Evol Comput201418457760110.1109/TEVC.2013.2281535 SaravananRRamabalanSEbenezerNGRDharmarajaCEvolutionary multi criteria design optimization of robot grippersAppl Soft Comput20099115917210.1016/j.asoc.2008.04.001 J Derrac (3794_CR15) 2011; 1 S Huband (3794_CR24) 2006; 10 S Jiang (3794_CR26) 2015; 45 S Holm (3794_CR21) 1979; 6 X Cai (3794_CR6) 2015; 19 CA Coello Coello (3794_CR8) 2002; 191 X Cai (3794_CR5) 2013; 17 OJ Dunn (3794_CR16) 1961; 56 3794_CR17 Y Hochberg (3794_CR19) 1988; 75 3794_CR39 3794_CR36 3794_CR37 MA Jan (3794_CR25) 2013; 13 K Miettinen (3794_CR31) 1999 PA Bosman (3794_CR4) 2003; 7 3794_CR11 E Zitzler (3794_CR41) 1999; 3 H Li (3794_CR28) 2009; 13 M Črepinšek (3794_CR10) 2013; 45 BS Holland (3794_CR20) 1987; 43 G Hommel (3794_CR22) 1988; 75 R Saravanan (3794_CR35) 2009; 9 X Cai (3794_CR7) 2017; 47 Z Hu (3794_CR23) 2013; 18 J Bader (3794_CR2) 2011; 19 K Deb (3794_CR14) 2002; 6 H Finner (3794_CR18) 1993; 88 TP Runarsson (3794_CR33) 2000; 4 HL Liu (3794_CR29) 2014; 18 3794_CR42 K Deb (3794_CR13) 2014; 18 Q Zhang (3794_CR38) 2007; 11 N Beume (3794_CR3) 2007; 181 E Mezura-Montes (3794_CR30) 2011; 1 3794_CR9 K Deb (3794_CR12) 2001 JD Li (3794_CR27) 2008; 138 3794_CR40 TP Runarsson (3794_CR34) 2005; 35 3794_CR1 DM Rom (3794_CR32) 1990; 77
References_xml	– reference: BeumeNNaujoksBEmmerichMSMS-EMOA: multiobjective selection based on dominated hypervolumeEur J Oper Res200718131653166910.1016/j.ejor.2006.08.008 – reference: Fan Z, Li W, Cai X, Li H, Hu K, Zhang Q, Deb K, Goodman ED (2016) Difficulty adjustable and scalable constrained multi-objective test problem toolkit. arXiv preprint arXiv:1612.07603 – reference: JiangSZhangJOngYSZhangANTanPSA simple and fast hypervolume indicator-based multiobjective evolutionary algorithmIEEE Trans Cybern201545102202221310.1109/TCYB.2014.2367526 – reference: DerracJGarcíaSMolinaDHerreraFA practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithmsSwarm Evol Comput20111131810.1016/j.swevo.2011.02.002 – reference: FinnerHOn a monotonicity problem in step-down multiple test proceduresJ Am Stat Assoc199388423920923124294210.1080/01621459.1993.10476358 – reference: Corne DW, Jerram NR, Knowles JD, Oates MJ (2001) PESA-II: region-based selection in evolutionary multiobjective optimization. In: Proceedings of the 3rd annual conference on genetic and evolutionary computation. Morgan Kaufmann, pp 283–290 – reference: Asafuddoula M, Ray T, Sarker R, Alam K (2012) An adaptive constraint handling approach embedded MOEA/D. In: 2012 IEEE Congress on Evolutionary Computation. IEEE, pp 1–8 – reference: DebKMulti-objective optimization using evolutionary algorithms2001LondonWiley0970.90091 – reference: Yang Z, Cai X, Fan Z (2014) Epsilon constrained method for constrained multiobjective optimization problems: some preliminary results. In: Proceedings of the Companion Publication of the 2014 Annual Conference on Genetic and Evolutionary Computation, ACM, pp 1181–1186 – reference: Coello CoelloCATheoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the artComput Methods Appl Mech Eng200219111–1212451287187768410.1016/S0045-7825(01)00323-1 – reference: BaderJZitzlerEHypE: an algorithm for fast hypervolume-based many-objective optimizationEvol Comput2011191457610.1162/EVCO_a_00009 – reference: JanMAKhanumRAA study of two penalty-parameterless constraint handling techniques in the framework of MOEA/DAppl Soft Comput201313112814810.1016/j.asoc.2012.07.027 – reference: RunarssonTPYaoXSearch biases in constrained evolutionary optimizationIEEE Trans Syst Man Cybern Part C Appl Rev200535223324310.1109/TSMCC.2004.841906 – reference: HollandBSCopenhaverMDAn improved sequentially rejective Bonferroni test procedureBiometrics19874341742389741010.2307/2531823 – reference: CaiXYangZFanZZhangQDecomposition-based-sorting and angle-based-selection for evolutionary multiobjective and many-objective optimizationIEEE Trans Cybern20174792824283710.1109/TCYB.2016.2586191 – reference: BosmanPAThierensDThe balance between proximity and diversity in multiobjective evolutionary algorithmsIEEE Trans Evol Comput20037217418810.1109/TEVC.2003.810761 – reference: ČrepinšekMLiuSHMernikMExploration and exploitation in evolutionary algorithms: a surveyACM Comput Surv20134533510.1145/2480741.2480752 – reference: ZitzlerEThieleLMultiobjective evolutionary algorithms: a comparative case study and the strength Pareto approachIEEE Trans Evol Comput19993425727110.1109/4235.797969 – reference: CaiXLiYFanZZhangQAn external archive guided multiobjective evolutionary algorithm based on decomposition for combinatorial optimizationIEEE Trans Evol Comput201519450852310.1109/TEVC.2014.2350995 – reference: LiuHLGuFZhangQDecomposition of a multiobjective optimization problem into a number of simple multiobjective subproblemsIEEE Trans Evol Comput201418345045510.1109/TEVC.2013.2281533 – reference: RunarssonTPYaoXStochastic ranking for constrained evolutionary optimizationIEEE Trans Evol Comput20004328429410.1109/4235.873238 – reference: DebKPratapAAgarwalSMeyarivanTA fast and elitist multiobjective genetic algorithm: NSGA-IIIEEE Trans Evol Comput20026218219710.1109/4235.996017 – reference: RomDMA sequentially rejective test procedure based on a modified Bonferroni inequalityBiometrika1990773663665108786010.1093/biomet/77.3.663 – reference: HubandSHingstonPBaroneLWhileLA review of multiobjective test problems and a scalable test problem toolkitIEEE Trans Evol Comput200610547750610.1109/TEVC.2005.861417 – reference: LiJDA two-step rejection procedure for testing multiple hypothesesJ Stat Plann Inference2008138615211527242728710.1016/j.jspi.2007.04.032 – reference: MiettinenKNonlinear multiobjective optimization1999BerlinSpringer0949.90082 – reference: Takahama T, Sakai S (2006) Constrained optimization by the ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} constrained differential evolution with gradient-based mutation and feasible elites. In: 2006 IEEE international conference on evolutionary computation. IEEE, pp 1–8 – reference: DebKJainHAn evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraintsIEEE Trans Evol Comput201418457760110.1109/TEVC.2013.2281535 – reference: CaiXHuZFanZA novel memetic algorithm based on invasive weed optimization and differential evolution for constrained optimizationSoft Comput201317101893191010.1007/s00500-013-1028-4 – reference: DunnOJMultiple comparisons among meansJ Am Stat Assoc196156293526412495210.1080/01621459.1961.10482090 – reference: Zitzler E, Künzli S (2004) Indicator-based selection in multiobjective search. In: International conference on parallel problem solving from nature. Springer, pp 832–842 – reference: ZhangQLiHMOEA/D: a multiobjective evolutionary algorithm based on decompositionIEEE Trans Evol Comput200711671273110.1109/TEVC.2007.892759 – reference: HommelGA stagewise rejective multiple test procedure based on a modified Bonferroni testBiometrika198875238338610.1093/biomet/75.2.383 – reference: LiHZhangQMultiobjective optimization problems with complicated pareto sets, MOEA/D and NSGA-IIIEEE Trans Evol Comput200913228430210.1109/TEVC.2008.925798 – reference: SaravananRRamabalanSEbenezerNGRDharmarajaCEvolutionary multi criteria design optimization of robot grippersAppl Soft Comput20099115917210.1016/j.asoc.2008.04.001 – reference: Zhang Q, Zhou A, Zhao S, Suganthan PN, Liu W, Tiwari S (2008) Multiobjective optimization test instances for the CEC 2009 special session and competition. University of Essex, Colchester, UK and Nanyang Technological University, Singapore, special session on performance assessment of multi-objective optimization algorithms, Technical report 264 – reference: HuZCaiXFanZAn improved memetic algorithm using ring neighborhood topology for constrained optimizationSoft Comput201318102023204110.1007/s00500-013-1183-7 – reference: Zitzler E, Laumanns M, Thiele L (2001) SPEA2: Improving the strength Pareto evolutionary algorithm. TIK-report 103 – reference: HochbergYA sharper Bonferroni procedure for multiple tests of significanceBiometrika198875480080299512610.1093/biomet/75.4.800 – reference: Mezura-MontesECoello CoelloCAConstraint-handling in nature-inspired numerical optimization: past, present and futureSwarm Evol Comput20111417319410.1016/j.swevo.2011.10.001 – reference: Datta R, Deb K (2011) Multi-objective design and analysis of robot gripper configurations using an evolutionary-classical approach. 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Snippet	This paper proposes an improved epsilon constraint-handling mechanism and combines it with a decomposition-based multi-objective evolutionary algorithm...
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StartPage	12491
SubjectTerms	Artificial Intelligence Collaboration Computational Intelligence Constraints Control Decomposition Engineering Evolutionary algorithms Mathematical Logic and Foundations Mechatronics Methodologies and Application Methods Multiple objective analysis Optimization Performance evaluation Robotics Violations
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Title	An improved epsilon constraint-handling method in MOEA/D for CMOPs with large infeasible regions
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