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An efficient solution strategy for bilevel multiobjective optimization problems using multiobjective evolutionary algorithm

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Abstract

An efficient solution strategy is proposed for bilevel multiobjective optimization problem (BLMOP) with multiple objectives at both levels when multiobjective optimization problem (MOP) at the lower level satisfies the convexity and differentiability for the lower-level decision variables. In the proposed strategy, the MOP at the lower level is first converted into a single-objective optimization formulation through adopting adaptive weighted sum scalarization, in which the lower-level weight vector is adjusted adaptively while the iteration progressing. The Karush-Kuhn-Tucker optimality conditions are used to the lower-level single-objective scalarization problem, thus the original BLMOP can be converted into a single-level MOP with complementarity constraints. Then an effective smoothing technique is suggested to cope with the complementarity constraints. In such a way, the BLMOP is finally formalized as a single-level constrained nonlinear MOP. A decomposition-based constrained multiobjective differential evolution is developed to solve this transformed MOP and some instances are tested to illustrate the feasibility and effectiveness of the solution methodology. The experimental results show that the proposed solution method possesses favorite convergence and diversity.

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References

  • Alves MJ, Dempe S, Júdice JJ (2012) Computing the Pareto frontier of a bi-objective bi-level linear problem using a multiobjective mixed-integer programming algorithm. Optimization 61(3):335–358

    Article  MathSciNet  MATH  Google Scholar 

  • Alves MJ, Costa JP (2014) An algorithm based on particle swarm optimization for multiobjective bilevel linear problems. Appl Math Comput 247:547–561

    MathSciNet  MATH  Google Scholar 

  • Angelo JS, Krempser E, Barbosa HJC (2013) Differential evolution for bilevel programming. In 2013 IEEE congress on evolutionary computation (CEC), pp. 470-477

  • Ankhili Z, Mansouri A (2009) An exact penalty on bilevel programs with linear vector optimization lower level. Eur J Oper Res 197:36–41

    Article  MathSciNet  MATH  Google Scholar 

  • Bard JF (1998) Practical bilevel optimization: algorithms and applications. Kluwer Academic Publishers, Dordrecht, The Netherlands

    Book  MATH  Google Scholar 

  • Bonnel H, Morgan J (2006) Semivectorial bilevel optimization problem: penalty approach. J Optimiz Theory App 131(3):365–382

    Article  MathSciNet  MATH  Google Scholar 

  • Bosman PAN, Thierens D (2003) The balance between proximity and diversity in multiobjective evolutionary algorithms. IEEE Trans Evol Comput 7(2):174–188

    Article  Google Scholar 

  • Calvete HI, Galé C (2010) Linear bilevel programs with multiple objectives at the upper level. J Comput Appl Math 234(4):950–959

    Article  MathSciNet  MATH  Google Scholar 

  • Calvete HI, Galé C (2011) On linear bilevel problems with multiple objectives at the lower level. Omega 39:33–40

    Article  Google Scholar 

  • Coello CAC (2000) An updated survey of GA-based multiobjective optimization techniques. ACM Comput Surv 32(2):109–143

    Article  Google Scholar 

  • Colson B, Marcotte P, Savard G (2005) Bilevel programming: a survey. 4OR 3(2):87–107

    Article  MathSciNet  MATH  Google Scholar 

  • Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153(1):235–256

    Article  MathSciNet  MATH  Google Scholar 

  • Das S, Suganthan PN (2011) Differential evolution: a survey of the state-of-the-art. IEEE Trans Evol Comput 15(1):4–31

    Article  Google Scholar 

  • Deb K, Agrawal S, Pratap A, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197

    Article  Google Scholar 

  • Deb K, Sinha A (2009a) Constructing test problems for bilevel evolutionary multi-objective optimization. IEEE congress on evolutionary computation, CEC’09, pp. 1153-1160

  • Deb K, Sinha A (2009b) Solving bilevel multi-objective optimization problems using evolutionary algorithms. In Evolutionary multi-criterion optimization. 5th international conference, EMO 2009, M. Ehrgott, C.M. Fonseca, X. Gandibleux, J.-K. Hao and M. Sevaux, eds, Lecture notes in computer science, vol. 5467, Springer-Verlag, Berlin, pp. 110-124

  • Deb K, Sinha A (2010) An efficient and accurate solution methodology for bilevel multiobjective programming problems using a hybrid evolutionary-local-search algorithm. Evol Comput 18(3):403–449

    Article  Google Scholar 

  • Dempe S (2002) Foundations of bilevel programming. Kluwer Academic Publishers, Dordrecht, The Netherlands

    MATH  Google Scholar 

  • Dempe S (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3):333–359

    Article  MathSciNet  MATH  Google Scholar 

  • Dempe S, Gadhi N, Zemkoho AB (2013) New optimality conditions for the semivectorial bilevel optimization problem. J Optimiz Theory App 157(1):54–74

    Article  MathSciNet  MATH  Google Scholar 

  • Eichfelder G (2010) Multiobjective bilevel optimization. Math Progr 123:419–449

    Article  MathSciNet  MATH  Google Scholar 

  • Facchinei F, Jiang H, Qi L (1999) A smoothing method for mathematical programs with equilibrium constraints. Math Progr 85:107–134

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta A, Ong YS, (2015) An evolutionary algorithm with adaptive scalarization for multiobjective bilevel programs. IEEE CEC, (2015) 25–28. Sendai, Japan, pp 1636–1642

  • Hejazi SR, Memariani A, Jahanshahloo G, Sepehri MM (2002) Linear bilevel programming solution by genetic algorithm. Comput Oper Res 29:1913–1925

    Article  MathSciNet  MATH  Google Scholar 

  • Jia L, Wang Y (2009) A genetic algorithm for multiobjective bilevel convex optimization problems. Int Conf Comput Intell Secur 1:98–102

    Google Scholar 

  • Leung YW, Wang Y (2000) Multiobjective programming using uniform design and genetic algorithm. IEEE Trans Syst Man Cybern C 30(3):293–304

    Article  Google Scholar 

  • Li X, Tian P, Min X (2006) A hierarchical particle swarm optimization for solving bilevel programming problems. Artificial intelligence and soft computing-ICAISC 2006. Lecture Notes Comput Sci 4029:1169–1178

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Li H, Jiao YC, Zhang FS, Zhang L (2009) An efficient method for linear bilevel programming problems based on the orthogonal genetic algorithm. Int J Innov Comp Inf Control 5:2837–2846

    Google Scholar 

  • Li H, Zhang L (2014) A differential evolution with two mutation strategies and a selection based on an improved constraint-handling technique for bilevel programming problems. Math Probl Eng 2014:1–16

    MathSciNet  Google Scholar 

  • Li H, Zhang L, Jiao YC (2016) An interactive approach based on a discrete differential evolution algorithm for a class of integer bilevel programming problems. Int J Syst Sci 47(10):2330–2341

    Article  MathSciNet  MATH  Google Scholar 

  • Li H, Zhang Q, Chen Q, Zhang L, Jiao YC (2016) Multiobjective differential evolution algorithm based on decomposition for a type of multiobjective bilevel programming problems. Knowl-Based Syst 107:271–288

    Article  Google Scholar 

  • Liu B (1998) Stackelberg-nash equilibrium for multilevel programming with multiple followers using genetic algorithms. Comput Math Appl 36(7):79–89

    Article  MathSciNet  MATH  Google Scholar 

  • Lu J, Han J, Hu Y, Zhang G (2016) Multilevel decision-making: a survey. Inf Sci 346:463–487

    Article  MathSciNet  MATH  Google Scholar 

  • Lu H, Yen GG (2003) Rank-density-based multiobjective genetic algorithm and benchmark test function study. IEEE Trans Evol Comput 7(4):325–343

    Article  Google Scholar 

  • Nishizaki I, Sakawa M (1999) Stakelberg solutions to multiobjective two-level linear programming problems. J Optimiz Theory App 103:161–182

    Article  MATH  Google Scholar 

  • Oduguwa V, Roy R (2002) Bi-level optimization using genetic algorithm. In Proc. IEEE Int. Conf. Artificial Intelligence Systems, pp. 123-128

  • Schott J (1995) Fault tolerant design using single and multicriteria genetic algorithm optimization. Master Thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA

  • Srinivas N, Deb K (1994) Multiobjective optimization using nondominated sorting in genetic algorithms. Evol Comput 2(3):221–248

    Article  Google Scholar 

  • Storn R, Price K (1997) Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359

    Article  MathSciNet  MATH  Google Scholar 

  • Tan KC, Lee TH, Khor EF (2001) Evolutionary algorithm with dynamic population size and local exploration for multiobjective optimization. IEEE Trans Evol Comput 5(6):565–588

    Article  Google Scholar 

  • Tan YY, Jiao YC, Li H, Wang XK (2012) MOEA/D-SQA: a multi-objective memetic algorithm based on decomposition. Eng Optimiz 44(9):1095–1115

    Article  Google Scholar 

  • Tan YY, Jiao YC, Li H, Wang XK (2012) A modification to MOEA/D-DE for multiobjective optimization problems with complicated Pareto sets. Inf Sci 213:14–38

    Article  MathSciNet  MATH  Google Scholar 

  • Tan YY, Jiao YC, Li H, Wang XK (2013) MOEA/D+uniform design: a new version of MOEA/D for optimization problems with many objectives. Comput Oper Res 40(6):1648–1660

    Article  MathSciNet  MATH  Google Scholar 

  • Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: a bibliography review. J Glob Optim 5(3):291–306

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Y, Jiao YC, Li H (2005) An evolutionary algorithm for solving nonlinear bilevel programming based on a new constraint-handing scheme. IEEE Trans Syst Man Cybern C 35(2):221–232

    Article  Google Scholar 

  • Wang Y, Li H, Dang C (2011) A new evolutionary algorithm for a class of nonlinear bilevel programming problems and its global convergence. INFORMS J Comput 23(4):618–629

    Article  MathSciNet  MATH  Google Scholar 

  • Wang JYT, Ehrgott M, Dirks KN, Gupta A (2014) A bilevel multi-objective road pricing model for economic, environmental and health sustainability. Transp Res Procedia 3:393–402

    Article  Google Scholar 

  • Ye JJ (2011) Necessary optimality conditions for multiobjective bilevel programs. Math Oper Res 36(1):165–184

    Article  MathSciNet  MATH  Google Scholar 

  • Yin Y (2002) Multiobjective bilevel optimization for transportation planning and management problems. J Adv Transp 36:93–105

    Article  Google Scholar 

  • Zhang G, Lu J, Dillon T (2007) Decentralized multi-objective bilevel decision making with fuzzy demands. Knowl-Based Syst 20(5):495–507

  • Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731

    Article  Google Scholar 

  • Zhang Q, Zhou A, Jin Y (2008) RM-MEDA: a regularity model-based multiobjective estimation of distribution algorithm. IEEE Trans Evol Comput 21(1):41–63

    Article  Google Scholar 

  • Zhang T, Hu T, Guo X, Chen Z, Zheng Y (2013) Solving high dimensional bilevel multiobjective programming problem using a hybrid particle swarm optimization algorithm with crossover operator. Knowl-Based Syst 53:13–19

    Article  Google Scholar 

  • Zhou A, Qu BY, Li H, Zhao SZ, Suganthan PN, Zhang Q (2011) Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evol Comput 1:32–49

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Zitzler E, Thiele L, Laumanns M, Fonseca CM, da Fonseca VG (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2):117–132

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank anonymous reviewers and the associate editor for their valuable comments and suggestions that greatly improved this paper’s quality.

Funding

This work was supported by the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2019JM-503), and the National Natural Science Foundation of China (Grant No. 61966030).

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  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    Hong Li & Li Zhang

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  1. Hong Li
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Li, H., Zhang, L. An efficient solution strategy for bilevel multiobjective optimization problems using multiobjective evolutionary algorithm. Soft Comput 25, 8241–8261 (2021). https://doi.org/10.1007/s00500-021-05750-0

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