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A Differential Evolution Algorithm to Semivectorial Bilevel Problems

  • Maria João AlvesEmail author
  • Carlos Henggeler Antunes
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10710)

Abstract

Semivectorial bilevel problems (SVBLP) deal with the optimization of a single function at the upper level and multiple objective functions at the lower level of hierarchical decisions. Therefore, a set of nondominated solutions to the lower level decision maker (the follower) exists and should be exploited for each setting of decision variables controlled by the upper level decision maker (the leader). This paper presents a new algorithmic approach based on differential evolution to compute a set of four extreme solutions to the SVBLP. These solutions capture not just the optimistic vs. pessimistic leader’s attitude but also possible follower’s reactions more or less favorable to the leader within the lower level nondominated solution set. The differential evolution approach is compared with a particle swarm optimization algorithm. In this experimental comparison we draw attention to pitfalls associated with the interpretation of results and assessment of the performance of algorithms in SVBLP.

Keywords

Semivectorial bilevel problems Differential evolution Particle swarm optimization Optimistic/pessimistic frontiers Optimistic/deceiving solutions Pessimistic/rewarding solutions 

Notes

Acknowledgment

This work was supported by projects UID/MULTI/00308/2013 and SAICTPAC/0004/2015-POCI-01-0145-FEDER-016434.

References

  1. 1.
    Alves, M.J., Antunes, C.H., Carrasqueira, P.: A PSO approach to semivectorial bilevel programming: pessimistic, optimistic and deceiving solutions. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2015), pp. 599–606 (2015)Google Scholar
  2. 2.
    Bonnel, H.: Optimality conditions for the semivectorial bilevel optimization problem. Pac. J. Optim. 2, 447–468 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bonnel, H., Morgan, J.: Semivectorial bilevel optimization problem: penalty approach. J. Optim. Theor. Appl. 131, 365–382 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ankhili, Z., Mansouri, A.: An exact penalty on bilevel programs with linear vector optimization lower level. Eur. J. Oper. Res. 197, 36–41 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Zheng, Y., Wan, Z.: A solution method for semivectorial bilevel programming problem via penalty method. J. Appl. Math. Comput. 37, 207–219 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ren, A., Wang, Y.: A novel penalty function method for semivectorial bilevel programming problem. Appl. Math. Model. 40, 135–149 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Calvete, H., Galé, C.: On linear bilevel problems with multiple objectives at the lower level. Omega 39, 33–40 (2011)CrossRefGoogle Scholar
  8. 8.
    Liu, B., Wan, Z., Chen, J., Wang, G.: Optimality conditions for pessimistic semivectorial bilevel programming problems. J. Inequal. Appl. 2014, 41 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lv, Y., Chen, J.: A discretization iteration approach for solving a class of semivectorial bilevel programming problem. J. Nonlinear Sci. Appl. 9, 2888–2899 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Alves, M.J., Antunes, C.H.: An illustration of different concepts of solutions in semivectorial bilevel programming. In: 2016 IEEE Symposium on Computational Intelligence (SSCI) (2016)Google Scholar
  11. 11.
    Mezura-Montes, E., Velázquez-Reyes, J., Coello Coello, C.A.: A comparative study of differential evolution variants for global optimization. In: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, pp. 485–492 (2006)Google Scholar
  12. 12.
    Deb, K., Sinha, A.: Solving bilevel multi-objective optimization problems using evolutionary algorithms. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, J.-K., Sevaux, M. (eds.) EMO 2009. LNCS, vol. 5467, pp. 110–124. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-01020-0_13CrossRefGoogle Scholar
  13. 13.
    Sinha, A., Malo, P., Deb, K.: Approximated set-valued mapping approach for handling multiobjective bilevel problems. Comput. Oper. Res. 77, 194–209 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Deb, K., Sinha, A.: Constructing test problems for bilevel evolutionary multi-objective optimization. In: 2009 IEEE Congress on Evolutionary Computation, pp. 1153–1160 (2009)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Maria João Alves
    • 1
    • 3
    Email author
  • Carlos Henggeler Antunes
    • 2
    • 3
  1. 1.CeBER and Faculty of EconomicsUniversity of CoimbraCoimbraPortugal
  2. 2.DEECUniversity of CoimbraCoimbraPortugal
  3. 3.INESC CoimbraCoimbraPortugal

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