Abstract
Evolutionary multi-objective optimization is a field that has experienced a rapid growth in the last two decades. Although an important number of new multi-objective evolutionary algorithms have been designed and implemented by the scientific community, the popular Non-Dominated Sorting Genetic Algorithm (NSGA-II) remains as a widely used baseline for algorithm performance comparison purposes and applied to different engineering problems. Since every evolutionary algorithm needs several parameters to be set up in order to operate, parameter control constitutes a crucial task for obtaining an effective and efficient performance in its execution. However, despite the advancements in parameter control for evolutionary algorithms, NSGA-II has been mainly used in the literature with fine-tuned static parameters. This paper introduces a novel and computationally lightweight self-adaptation mechanism for controlling the distribution index parameter of the polynomial mutation operator usually employed by NSGA-II in particular and by multi-objective evolutionary algorithms in general. Additionally, the classical NSGA-II using polynomial mutation with a static distribution index is compared with this new version utilizing a self-adapted parameter. The experiments carried out over twenty-five benchmark problems show that the proposed modified NSGA-II with a self-adaptive mutator outperforms its static counterpart in more than 75% of the problems using three quality metrics (hypervolume, generalized spread, and modified inverted generational distance).
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Notes
A deceptive search space is characterized by the fact that most of it tends to guide the search towards areas which are far from the global optimum, thus leading to a suboptimal local optimum.
References
Aleti A, Moser I (2016) A systematic literature review of adaptive parameter control methods for evolutionary algorithms. ACM Comput Surv (CSUR) 49(3):1–35. https://doi.org/10.1145/2996355
Angeline PJ (1995) Adaptive and self-adaptive evolutionary computations. In: Palaniswami M, Attikiouzel Y (eds) Computational intelligence: a dynamic systems perspective. IEEE Press, New York, pp 152–163
Audet C, Bigeon J, Cartier D, Digabel SL, Salomon L (2020) Performance indicators in multiobjective optimization. Eur J Oper Res 292(2):397–422. https://doi.org/10.1016/j.ejor.2020.11.016
Auger A, Bader J, Brockhoff D, Zitzler E (2009) Theory of the hypervolume indicator: optimal \(\mu \)-distributions and the choice of the reference point. In: Proceedings of the tenth ACM SIGEVO workshop on foundations of genetic algorithms - FOGA ’09 https://doi.org/10.1145/1527125.1527138
Auger A, Stutzle T, Sharma M, Komninos A, López-Ibánez M, Kazakov D (2019) Deep reinforcement learning based parameter control in differential evolution. In: Proceedings of the genetic and evolutionary computation conference pp. 709–717. https://doi.org/10.1145/3321707.3321813
Back T (1992) The interaction of mutation rate, selection, and self-adaptation within a genetic algorithm. In: Parallel problem solving from nature 2, PPSN-II. Elsevier, Brussels, Belgium
Back T, Schutz M (1996) Intelligent mutation rate control in canonical genetic algorithms. In: ISMIS ’96: Proceedings of the 9th international symposium on foundations of intelligent systems. Springer, Berlin, Heidelberg, pp 158–167, https://doi.org/10.1007/3-540-61286-6_141
Bader J, Zitzler E (2011) HypE: an algorithm for fast hypervolume-based many-objective optimization. Evol Comput 19(1):45–76. https://doi.org/10.1162/evco_a_00009
Birattari M, Yuan Z, Balaprakash P, Stuzle T (2010) F-race and iterated f-race: an overview. In: Experimental methods for the analysis of optimization algorithms. Springer, Berlin, Heidelberg, pp 311–336, https://doi.org/10.1007/978-3-642-02538-9_13
Bosman PAN, Thierens D (2003) The balance between proximity and diversity in multiobjective evolutionary algorithms. IEEE Trans Evol Comput. https://doi.org/10.1109/tevc.2003.810761
Bosman PAN, Cruz-Salinas AF, Perdomo JG (2017) Self-adaptation of genetic operators through genetic programming techniques. In: Proceedings of the genetic and evolutionary computation conference. Association for Computing Machinery, Berlin, pp 913–920, https://doi.org/10.1145/3071178.3071214
Coello CAC, Sierra MR (2004) A study of the parallelization of a coevolutionary multi-objective evolutionary algorithm. In: MICAI 2004: advances in artificial intelligence, third Mexican international conference on artificial intelligence. Springer, Mexico City, https://doi.org/10.1007/978-3-540-24694-7_71
Deb K, Agrawal S, Pratap A, Mayarivan T (2000) A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. In: Parallel problem solving from nature PPSN VI, lecture notes in computer science, vol 1917. Springer, Berlin, pp 849–858, https://doi.org/10.1007/3-540-45356-3_83
Deb K, Thiele L, Laumanns M, Zitzler E (2002) Scalable multi-objective optimization test problems. In: Proceedings of the 2002 congress on evolutionary computation. CEC’02, vol 1. IEEE, Honolulu, pp 825–830, https://doi.org/10.1109/cec.2002.1007032
Deb K, Sindhya K, Okabe T (2007) Self-adaptive simulated binary crossover for real-parameter optimization. In: Proceedings of the 9th annual conference on genetic and evolutionary computation. Association for computing machinery, London, GECCO ’07, p 1187-1194, https://doi.org/10.1145/1276958.1277190
Doerr B, Doerr C (2020) Theory of parameter control for discrete black-box optimization: provable performance gains through dynamic parameter choices. In: Theory of evolutionary computation, recent developments in discrete optimization. Springer International Publishing, Cham, pp 271–321, https://doi.org/10.1007/978-3-030-29414-4_6
Durillo JJ, Nebro AJ (2011) jMetal: a Java framework for multi-objective optimization. In: Advances in Engineering Software, vol 42, no. 10. Elsevier, Oxford, pp 760–771, https://doi.org/10.1016/j.advengsoft.2011.05.014
Eiben A, Hinterding R, Michalewicz Z (1999) Parameter control in evolutionary algorithms. IEEE Trans Evol Comput 3(2):124–141. https://doi.org/10.1109/4235.771166
Eiben AE, Horvath M, Kowalczyk W, Schut MC (2007) Reinforcement learning for online control of evolutionary algorithms. In: Engineering self-organising systems, 4th international workshop, ESOA 2006. Springer, Hakodate, Japan, pp 151–160, https://doi.org/10.1007/978-3-540-69868-5_10
Garg H (2019) A hybrid GSA-GA algorithm for constrained optimization problems. Inf Sci 478:499–523. https://doi.org/10.1016/j.ins.2018.11.041
Grefenstette JJ (1986) Optimization of control parameters for genetic algorithms. IEEE Trans Syst Man Cybern 16(1):122–128. https://doi.org/10.1109/tsmc.1986.289288
Hamdan MM (2012) The distribution index in polynomial mutation for evolutionary multiobjective optimisation algorithms: an experimental study. In: Proceedings of international conference on electronics computer technology
Hamdan MM (2014) Revisiting the distribution index in simulated binary crossover operator for evolutionary multiobjective optimisation algorithms. In: 2014 fourth international conference on digital information and communication technology and its applications (DICTAP) pp 37–41. https://doi.org/10.1109/dictap.2014.6821653
Hansen MP, Jaszkiewicz A (1998) Evaluating the quality of approximations to the non-dominated set. Technical University of Denmark, Technical Report IMM-REP-1998-7, Denmark
Hassanat A, Almohammadi K, Alkafaween E, Abunawas E, Hammouri A, Prasath VBS (2019) Choosing mutation and crossover ratios for genetic algorithms-a review with a new dynamic approach. Information 10(12):390. https://doi.org/10.3390/info10120390
Hinterding R, Michalewicz Z, Eiben AE (1997) Adaptation in evolutionary computation: a survey. In: Proceedings of 1997 IEEE international conference on evolutionary computation (ICEC ’97). IEEE, Indianapolis
Huang C, Li Y, Yao X (2020) A survey of automatic parameter tuning methods for metaheuristics. IEEE Trans Evol Comput 24(2):201–216. https://doi.org/10.1109/tevc.2019.2921598
Huang C, Bai H, Yao X (2022) Online algorithm configuration for differential evolution algorithm. Appl Intell. https://doi.org/10.1007/s10489-021-02752-1
Huband S, Barone L, While L, Hingston P (2005) A scalable multi-objective test problem toolkit. In: Evolutionary multi-criterion optimization, third international conference, EMO 2005, lecture notes in computer science, vol 3410. Springer, Guanajuato, México, pp 280–295, https://doi.org/10.1007/978-3-540-31880-4_20
Ishibuchi H, Yoshida T, Murata T (2003) Balance between genetic search and localsearch in memetic algorithms for multiobjective permutation flowshop scheduling. IEEE Trans Evol Comput 7(2):204–223. https://doi.org/10.1109/tevc.2003.810752
Ishibuchi H, Masuda H, Tanigaki Y, Nojima Y (2015) Modified distance calculation in generational distance and inverted generational distance. In: Proceedings of 8th international conference on evolutionary multi-criterion optimization. Springer, Guimaraes, Portugal, pp 110–125, https://doi.org/10.1007/978-3-319-15892-1_8
Ishibuchi H, Imada R, Setoguchi Y, Nojima Y (2018) How to specify a reference point in hypervolume calculation for fair performance comparison. Evol Comput 26(3):411–440. https://doi.org/10.1162/evco_a_00226
Jiang S, Ong YS, Zhang J, Feng L (2014) Consistencies and contradictions of performance metrics in multiobjective optimization. IEEE Trans Cybern 44(12):2391–2404. https://doi.org/10.1109/tcyb.2014.2307319
Jong KAD (1975) Analysis of the beavior of a class of genetic adaptive systems. PhD thesis, Computer and Communication Sciences Department, University of Michigan
Karafotias G, Hoogendoorn M, Eiben AE (2015) Parameter control in evolutionary algorithms: trends and challenges. IEEE Trans Evol Comput 19(2):167–187. https://doi.org/10.1109/tevc.2014.2308294
Knowles J, Corne D (2002) On metrics for comparing nondominated sets. In: Proceedings of the 2002 congress on evolutionary computation. CEC’02, vol 1. IEEE, Honolulu, pp 711–716, https://doi.org/10.1109/cec.2002.1007013
Knowles JD (2002) Local-search and hybrid evolutionary algorithms for pareto optimization. PhD thesis, Department of Computer Science, University of Reading
Kochenderfer MJ, Wheeler TA (2019) Algorithms for optimization. The MIT Press, Cambridge and London
Korejo I, Yang S, Li C (2009) A comparative study of adaptive mutation operators for genetic algorithms. In: The VIII metaheuristic international conference, Hamburg, Germany
Kundu T, Garg H (2022) A hybrid ITLHHO algorithm for numerical and engineering optimization problems. Int J Intell Syst 37(7):3900–3980. https://doi.org/10.1002/int.22707
Kundu T, Garg H (2022) LSMA-TLBO: a hybrid SMA-TLBO algorithm with lévy flight based mutation for numerical optimization and engineering design problems. Adv Eng Software. https://doi.org/10.1016/j.advengsoft.2022.103185
Kursawe F (1991) A variant of evolution strategies for vector optimization. In: Schwefel HP, Männer R (eds) Parallel problem solving from nature. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 193–197, https://doi.org/10.1007/BFb0029752
Lacerda MGPd, Pessoa LFdA, Neto FBdL, Ludermir TB, Kuchen H (2021) A systematic literature review on general parameter control for evolutionary and swarm-based algorithms. Swarm Evol Comput. https://doi.org/10.1016/j.swevo.2020.100777
Lee CY, Yao X (2004) Evolutionary programming using mutations based on the Lévy probability distribution. IEEE Trans Evol Comput 8(1):1–13. https://doi.org/10.1109/tevc.2003.816583
Li M, Yao X (2019) Quality evaluation of solution sets in multiobjective optimisation: a survey. ACM Comput Surv (CSUR) 52(2):26. https://doi.org/10.1145/3300148
Liu Z, Chen G, Ong C, Yao Z, Li X, Deng J, Cui F (2023) Multi-objective design optimization of stent-grafts for the aortic arch. Mater Des. https://doi.org/10.1016/j.matdes.2023.111748
Lobo FG, Lima CF, Michalewicz Z (2007) Parameter setting in evolutionary algorithms, studies in computation intelligence, vol 54. Springer, Berlin, Heidelberg,. https://doi.org/10.1007/978-3-540-69432-8
Long Q, Li G, Jiang L (2022) A novel solver for multi-objective optimization: dynamic non-dominated sorting genetic algorithm (DNSGA). Soft Comput 26(2):725–747. https://doi.org/10.1007/s00500-021-06223-0
Lopez EM, A C, Coello C (2016) IGD+ -EMOA: a multi-objective evolutionary algorithm based on IGD+. In: 2016 IEEE congress on evolutionary computation (CEC), pp 999–1006, https://doi.org/10.1109/cec.2016.7743898
López-Ibánez M, Dubois-Lacoste J, Cáceres LP, Birattari M, Stutzle T (2016) The irace package: iterated racing for automatic algorithm configuration. Oper Res Perspect 3:43–58. https://doi.org/10.1016/j.orp.2016.09.002
Mezura-Montes E, Palomeque-Ortiz AG (2009) Self-adaptive and deterministic parameter control in differential evolution for constrained optimization. In: Constraint-handling in evolutionary optimization, pp 95–120, https://doi.org/10.1007/978-3-642-00619-7_5
Mohamed A, Oliva D, Suganthan P (2022) Handbook of nature-inspired optimization algorithms: the state of the art: Volume II: solving constrained single objective real-parameter optimization problems. Studies in systems, decision and control, Springer International Publishing https://doi.org/10.1007/978-3-031-07516-2
Nama S, Sharma S, Saha AK, Gandomi AH (2022) A quantum mutation-based backtracking search algorithm. Artif Intell Rev 55(4):3019–3073. https://doi.org/10.1007/s10462-021-10078-0
Nebro AJ, Luna F, Alba E, Dorronsoro B, Durillo JJ, Beham A (2008) AbYSS: adapting scatter search to multiobjective optimization. IEEE Trans Evol Comput 12(4):439–457. https://doi.org/10.1109/tevc.2007.913109
Okabe T, Jin Y, Sendhoff B (2003) A critical survey of performance indices for multi-objective optimisation. In: The 2003 congress on evolutionary computation, 2003. CEC ’03, vol 2. IEEE, Canberra, Australia, pp 878–885, https://doi.org/10.1109/cec.2003.1299759
Ozcelikkan N, Tuzkaya G, Alabas-Uslu C, Sennaroglu B (2022) A multi-objective agile project planning model and a comparative meta-heuristic approach. Inf Softw Technol. https://doi.org/10.1016/j.infsof.2022.107023
Papa G, (2021) Applications of dynamic parameter control in evolutionary computation. In, (2021) Genetic and evolutionary computation conference companion (GECCO ’21 Companion). ACM, Lille, France, proceedings of the genetic and evolutionary computation conference companion, DOI 10(1145/3449726):3461435
Parpinelli RS, Plichoski GF, Silva RSD, Narloch PH (2019) A review of techniques for online control of parameters in swarm intelligence and evolutionary computation algorithms. Int J Bio-Inspired Comput 13(1):1. https://doi.org/10.1504/ijbic.2019.097731
Rahimi I, Gandomi AH, Deb K, Chen F, Nikoo MR (2022) Scheduling by NSGA-II: review and bibliometric analysis. Processes 10(1):98. https://doi.org/10.3390/pr10010098
Rajabi A, Witt C (2020) Self-adjusting evolutionary algorithms for multimodal optimization. In: Proceedings of GECCO ’20. ACM Press, Cancun, Mexico, pp 1314–1322, https://doi.org/10.1007/s00453-022-00933-z
Rechenberg I (1971) Evolutionsstrategie; optimierung technischer systeme nach prinzipien der biologischen evolution. PhD thesis, Department of Process Engineering, Technical University of Berlin
Riquelme N, Lucken CV, Barán B, (2015) Performance metrics in multi-objective optimization. In: 2015 Latin American computing conference (CLEI). IEEE, Arequipa, Perú,. https://doi.org/10.1109/clei.2015.7360024
Schaffer JD (1985) Multiple objective optimization with vector evaluated genetic algorithms. In: Proceedings of the 1st international conference on genetic algorithms. L. Erlbaum Associates Inc., Sheffield, UK, pp 93–100
Schott JR (1995) Fault tolerant design using single and multicriteria genetic algorithm optimization. PhD thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology
Sharma S, Khodadadi N, Saha AK, Gharehchopogh FS, Mirjalili S (2023) Non-dominated sorting advanced butterfly optimization algorithm for multi-objective problems. J Bionic Eng 20(2):819–843. https://doi.org/10.1007/s42235-022-00288-9
Smith J, Fogarty T (1996) Self adaptation of mutation rates in a steady state genetic algorithm. In: Proceedings of 1996 IEEE international conference on evolutionary computation. IEEE, Nagoya, Japan, pp 318–323, https://doi.org/10.1109/icec.1996.542382
Smith JE, Fogarty TC (1997) Operator and parameter adaptation in genetic algorithms. Soft Comput 1(2):81–87. https://doi.org/10.1007/s005000050009
Srinivas N, Deb K (1994) Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol Comput 2(3):221–248. https://doi.org/10.1162/evco.1994.2.3.221
Storn R, Price K (1997) Differential evolution: a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11(4):341–359. https://doi.org/10.1023/a:1008202821328
Tan K, Chiam S, Mamun A, Goh C (2009) Balancing exploration and exploitation with adaptive variation for evolutionary multi-objective optimization. Eur J Oper Res 197(2):701–713. https://doi.org/10.1016/j.ejor.2008.07.025
Tanabe R, Ishibuchi H (2020) An Analysis of Quality Indicators Using Approximated Optimal Distributions in a 3-D Objective Space. IEEE Trans Evol Comput 24(5):853–867
Tanaka M, Watanabe H, Furukawa Y, Tanino T (1995) GA-based decision support system for multicriteria optimization. In: 1995 IEEE international conference on systems, man and cybernetics. Intelligent systems for the 21st century, vol 2. IEEE, Vancouver, British Columbia, Canada, pp 1556–1561, https://doi.org/10.1109/icsmc.1995.537993
Tinós R, Yang S (2007) Self-adaptation of mutation distribution in evolutionary algorithms. In: 2007 IEEE congress on evolutionary computation. IEEE, Singapore, pp 79–86, https://doi.org/10.1109/cec.2007.4424457
Veldhuizen DAV, Lamont GB (1998) Evolutionary computation and convergence to a pareto front. Late-breaking papers book at the genetic programming 1998 conference (GP-98). Stanford University Bookstore, Winsconsin, pp 221–228
Wang J, Liu Y, Ren S, Wang C, Ma S (2023) Edge computing-based real-time scheduling for digital twin flexible job shop with variable time window. Robot Comput Integr Manuf. https://doi.org/10.1016/j.rcim.2022.102435
Wang S, Ali S, Yue T, Li Y, Liaaen M (2016) A practical guide to select quality indicators for assessing pareto-based search algorithms in search based software engineering. In: IEEE/ACM 38th IEEE international conference on software engineering. IEEE, Austin, https://doi.org/10.1145/2884781.2884880
Yang S, Uyar S (2006) Adaptive mutation with fitness and allele distribution correlation for genetic algorithms. In: Proceedings of the 2006 ACM symposium on Applied computing - SAC ’06. ACM, Dijon, France, pp 940–944, https://doi.org/10.1145/1141277.1141499
Zeng F, Low MYH, Decraene J, Zhou S, Cai W (2010) Self-adaptive mechanism for multi-objective evolutionary algorithms. In: Proceedings of the international multiconference of engineers and computer scientists pp. 7–12
Zhang J, Chen WN, Zhan ZH, Yu WJ, Li YL, Chen N, Zhou Q (2012) A survey on algorithm adaptation in evolutionary computation. Front Electr Electr Eng 7(1):16–31. https://doi.org/10.1007/s11460-012-0192-0
Zhao Z, Liu B, Zhang C, Liu H (2019) An improved adaptive NSGA-II with multi-population algorithm. Appl Intell 49(2):569–580. https://doi.org/10.1007/s10489-018-1263-6
Zhou A, Jin Y, Zhang Q, Sendhoff B, Tsang E (2006) Combining model-based and genetics-based offspring generation for multi-objective optimization using a convergence criterion. In: 2006 IEEE international conference on evolutionary computation. IEEE, Vancouver, pp 892–899, https://doi.org/10.1109/cec.2006.1688406
Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evol Comput 3(4):257–271. https://doi.org/10.1109/4235.797969
Zitzler E, Deb K, Thiele L (1999) Comparison of multiobjective evolutionary algorithms: empirical results. Evol Comput 8(2):173–195. https://doi.org/10.1162/106365600568202
Zitzler E, Thiele L, Laumanns M, Fonseca CM, Fonseca VGd (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(13):117–132. https://doi.org/10.1109/tevc.2003.810758
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Carles-Bou, J.L., Galán, S.F. Self-adaptive polynomial mutation in NSGA-II. Soft Comput 27, 17711–17727 (2023). https://doi.org/10.1007/s00500-023-09049-0
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DOI: https://doi.org/10.1007/s00500-023-09049-0