Abstract
Multi-objective evolutionary algorithms (MOEAs) have shown their effectiveness in exploring a well converged and diversified approximation set for multi-objective optimization problems (MOPs) with 2 and 3 objectives. However, most of them perform poorly when tackling MOPs with more than 3 objectives [often called many-objective optimization problems (MaOPs)]. This is mainly due to the fact that the number of non-dominated individuals increases rapidly in MaOPs, leading to the loss of selection pressure in population update. Objective reduction can be used to lower the difficulties of some MaOPs, which helps to alleviate the above problem. This paper proposes a novel objective reduction framework for MaOPs using objective subspace extraction, named OSEOR. A new conflict information measurement among different objectives is defined to sort the relative importance of each objective, and then an effective approach is designed to extract several overlapped subspaces with reduced dimensionality during the execution of MOEAs. To validate the effectiveness of the proposed approach, it is embedded into a well-known and frequently used MOEA (NSGA-II). Several test MaOPs, including four irreducible problems (i.e. DTLZ1–DTLZ4) and a reducible problem (i.e. DTLZ5), are used to assess the optimization performance. The experimental results indicate that the performance of NSGA-II can be significantly enhanced using OSEOR on both irreducible and reducible MaOPs.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Aguirre H and Tanaka K (2009) Adaptive \(\varepsilon \)-ranking on MNK-landscapes. In: Computational intelligence in miulti-criteria decision-making, pp 104–111
Al MN, Petrovski A et al (2013) D2MOPSO: MOPSO based on decomposition and dominance with archiving using crowding distance in objective and solution spaces. Evol Comput 22(1):47–77
Auger A and Bader J, et al (2009) Theory of the hypervolume indicator: optimal \(\mu \)-distributions and the choice of the reference point. In: Tenth ACM Sigevo workshop on foundations of genetic algorithms, pp 87–102
Bader J, Zitzler E (2011) HypE: an algorithm for fast hypervolume-based many-objective optimization. Evol Comput 19(1):45–76
Brockhoff D, Zitzler E (2006) Are all objectives necessary? On dimensionality reduction in evolutionary multiobjective optimization. Lect Notes Comput Sci 4193(1):533–542
Brockhoff D, Zitzler E (2009) Objective reduction in evolutionary multiobjective optimization: theory and applications. Evol Comput 17(2):135–166
Chen J, Lin Q et al (2011) Chaos-based multi-objective immune algorithm with a fine-grained selection mechanism. Soft Comput 15(7):1273–1288
Chen XH, Li X et al (2014) Improved population partitioning method in multi-objective shuffled frog leaping algorithm. J Signal Process 30(10):1134–1142
Coello CAC, Lechuga MS (2002) MOPSO: a proposal for multiple objective particle swarm optimization. In: Proceedings of the 2002 congress on evolutionary computation. CEC’02, pp 1051–1056
Coello CAC (2005) Recent trends in evolutionary multiobjective optimization. Evolutionary multiobjective optimization. Springer, London
Coello CAC, Van Veldhuizen DA et al (2006) Evolutionary algorithms for solving multi-objective problems (Genetic and Evolutionary Computation). Springer, New York
Corne DW, Knowles JD (2009) Techniques for highly multiobjective optimisation: some nondominated points are better than others. In: Proceedings Gecco ACM, pp 773–780
Czyzżak P, Jaszkiewicz A (1998) Pareto simulated annealing: a metaheuristic technique for multiple-objective combinatorial optimization. J Multi Criteria Decis Anal 7(7):34–47
Deb K, Pratap A et al (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197
Deb K, Saxena DK (2005) On finding pareto-optimal solutions through dimensionality. Reduction for certain large-dimensional multi-objective optimization problems, KanGAL Report
Deb K, Thiele L et al (2006) Scalable test problems for evolutionary multiobjective optimization. Evolutionary Multiobjective Optimization, Springer, London, pp 105–145
Fonseca CM, Fleming PJ (1995) An overview of evolutionary algorithms in multiobjective optimization. Evol Comput 3(1):1–16
Gal T, Hanne T (1999) Consequences of dropping nonessential objectives for the application of MCDM methods. Eur J Oper Res 119(2):373–378
Giagkiozis I, Fleming PJ (2014) Pareto front estimation for decision making. Evol Comput 22(4):651–678
Hadka D, Reed PM et al (2012) Diagnostic assessment of the borg MOEA for many-objective product family design problems. Evolutionary computation, pp 1–10
Hadka D, Reed P (2011) Diagnostic assessment of search controls and failure modes in many-objective evolutionary optimization. Evol Comput 20(3):423–452
Huaping Y, Yao W (2013) A new multi-objective particle swarm optimization algorithm based on decomposition. J Syst Eng Electron 325(2):541–557
Ishibuchi H, Akedo N et al (2011) Behavior of EMO algorithms on many-objective optimization problems with correlated objectives. In: IEEE congress on evolutionary computation, Cec 2011, New Orleans, LA, USA, 5–8 June, pp 1465–1472
Ishibuchi H, Tsukamoto N et al (2008) Evolutionary many-objective optimization: a short review. In: Proc. of 2008 IEEE Congress on Evolutionary Computation, Hong Kong, pp 2424–2431
Ishibuchi H, Sakane Y et al (2009) Evolutionary many-objective optimization by NSGA-II and MOEA/D with large populations. In: IEEE international conference on systems, man and cybernetics, pp 1758–1763
Jaimes AL, Coello CAC et al (2008) Objective reduction using a feature selection technique. In: Genetic and evolutionary computation conference, GECCO 2008, Proceedings, Atlanta, GA, USA, pp 673–680
Jaimes AL, Coello CAC et al (2014) Objective space partitioning using conflict information for solving many-objective problems. Inf Sci 268(2):305–327
Jaimes AL, Coello CAC (2015) Many-objective problems: challenges and methods. Springer handbook of computational intelligence. Springer Berlin, Heidelberg, pp 1033–1046
Joshi R, Deshpande B (2014) Empirical and analytical study of many-objective optimization problems: analysing distribution of nondominated solutions and population size for scalability of randomized heuristics. Memet Comput 6(2):133–145
Li H, Landasilva D (2011) An adaptive evolutionary multi-objective approach based on simulated annealing. Evol Comput 19(4):561–595
Li H, Zhang Q (2009) Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II. IEEE Trans Evol Comput 13(2):284–302
Li X, Luo J et al (2012) An improved shuffled frog-leaping algorithm with extremal optimisation for continuous optimisation. Inf Sci 192(6):143–151
Lin Q, Chen J (2013) A novel micro-population immune multiobjective optimization algorithm. Comput Oper Res 40(6):1590–1601
Lin Q, Zhu Q et al (2015) A novel hybrid multi-objective immune algorithm with adaptive differential evolution. Comput Oper Res 62:95–111
Lin Q, Chen J et al (2016) A hybrid evolutionary immune algorithm for multiobjective optimization problems. IEEE Trans Evol Comput 20(5):711–729
Lotov AV, Bushenkov VA et al (2004) Introduction to interactive decision maps. Interactive decision maps. Springer, New York
Mashwani WK, Salhi A (2014) Multiobjective memetic algorithm based on decomposition. Appl Soft Comput 21(8):221–243
Obayashi S, Sasaki D (2003) Visualization and data mining of pareto solutions using self-organizing map. Int Conf Evo Multi Criterion Optim 2632:796–808
Parmee IC, Cvetkovic D (2002) Preferences and their application in evolutionary multiobjective optimisation. IEEE Trans Evol Comput 6(1):42–57
Powell WB (2011) Approximate dynamic programming: solving the curse of dimensionality, 2nd edn. Wiley, London
Saxena DK, Duro JX et al (2013) Objective reduction in many-objective optimization: linear and nonlinear algorithms. IEEE Trans Evol Comput 17(1):77–99
Saxena DK, Deb K (2007) Non-linear dimensionality reduction procedures for certain large-dimensional multi-objective optimization problems: employing correntropy and a novel maximum variance unfolding. Evol Multi Criter Optim 4403:772–787
Saxena D (2006) Searching for Pareto-optimal solutions through dimensionality reduction for certain large-dimensional multi-objective optimization problems. In: IEEE congress on evolutionary computation
Schaffer JD (1985) Multiple objective optimization with vector evaluated genetic algorithms. In: International conference on genetic algorithms, Pittsburgh, PA, USA, July, pp 93–100
Schott JR (1995) Fault tolerant design using single and multicriteria genetic algorithm optimization. Cell Immunol 37(1):1–13
Walker DJ, Everson R et al (2012) Visualizing mutually nondominating solution sets in many-objective optimization. IEEE Trans Evol Comput 17(2):165–184
Wang G, Wu JJ (2007) A new fuzzy dominance GA applied to solve many-objective optimization problem. In: International conference on innovative computing, information and control, IEEE computer society
Wang H, Jiao L et al (2015) A memetic optimization strategy based on dimension reduction in decision space. Evol Comput 23(1):69–100
Wang R, Purshouse RC et al (2013) Preference-inspired co-evolutionary algorithm using adaptively generated goal vectors. Evolutionary computation, pp 916–923
Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82
Zhan ZH, Li J et al (2013) Multiple populations for multiple objectives: a coevolutionary technique for solving multiobjective optimization problems. IEEE Trans Syst Man Cybernet Part B Cybernet Publ IEEE Syst Man Cybernet Soc 43(2):445–463
Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731
Zitzler E, Thiele L et al (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2):117–132
Zitzler E, Laumanns M et al (2001) SPEA2: improving the strength pareto evolutionary algorithm for multi-objective optimization. Evolutionary methods for design, optimization and control with applications to industrial problems. In: Proceedings of the Eurogen 2001, Athens, Greece, September
Zitzler E, Künzli S (2015) Indicator-based selection in multiobjective search. Lect Notes Comput Sci 3242:832–842
Zou X, Chen Y et al (2008) A new evolutionary algorithm for solving many-objective optimization problems. IEEE Trans Syst Man Cybernet Part B Cybernet Publ IEEE Syst Man Cybernet Soc 38(5):1402–1412
Acknowledgements
This work was supported by the National Nature Science Foundation of China under Grants 61402291, 61171124, 61301298, Seed Funding from Scientific and Technical Innovation Council of Shenzhen Government under Grant 0000012528, Foundation for Distinguished Young Talents in Higher Education of Guangdong under Grant 2014KQNCX129, Natural Science Foundation of SZU under Grants 201531, JCYJ20160422112909302, GJHS20160328145558586, and Science and Technology Planning Project of Guangdong under Grant 2013B021500017.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
Author Naili Luo declares that she has no conflict of interest. Author Xia Li declares that she has no conflict of interest. Author Qiuzhen Lin declares that he has no conflict of interest.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Communicated by A. Di Nola.
Rights and permissions
About this article
Cite this article
Luo, N., Li, X. & Lin, Q. Objective reduction for many-objective optimization problems using objective subspace extraction. Soft Comput 22, 1159–1173 (2018). https://doi.org/10.1007/s00500-017-2498-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-017-2498-6