New concepts and an algorithm for multiobjective bilevel programming: optimistic, pessimistic and moderate solutions

  • Maria João AlvesEmail author
  • Carlos Henggeler Antunes
  • João Paulo Costa
Original paper


Bilevel optimization deals with hierarchical mathematical programming problems in which two decision makers, the leader and the follower, control different sets of variables and have their own objective functions subject to interdependent constraints. Whenever multiple objective functions exist at the lower-level problem, the leader should cope with the uncertainty pertaining to the follower’s reaction. The leader can adopt a more optimistic or more pessimistic stance regarding the follower’s choice within his efficient region, which is restricted by the leader’s choice. Moreover, the leader may also have multiple objective functions. This paper presents new concepts associated with solutions to problems with multiple objective functions at the lower-level and a single or multiple objective functions at the upper-level, exploring the optimistic and pessimistic leader’s perspectives and their interplay with the follower’s choices. Extreme solutions (called optimistic/deceiving and pessimistic/rewarding) and a moderate solution, resulting from the risk the leader is willing to accept, are defined for problems with a single objective at the upper-level (semivectorial problems). Definitions of optimistic and pessimistic Pareto fronts are proposed for problems with multiple objective functions at the upper-level. These novel concepts are illustrated emphasizing the difficulties associated with the computation of those solutions. In addition, a differential evolution algorithm, approximating the extreme and moderate solutions for the semivectorial problem, is presented. Illustrative results of this algorithm further stress the challenges and pitfalls associated with the computation and interpretation of results in this kind of problems, which have not been properly addressed in literature and may lead to misleading conclusions.


Multiobjective bilevel optimization Semivectorial bilevel problem Optimistic and pessimistic Pareto fronts Optimistic, deceiving, pessimistic, rewarding and moderate solutions 

Mathematics Subject Classification

90B50 90C26 90C29 



This work was supported by projects UID/MULTI/00308/2013, ESGRIDS (POCI-01-0145-FEDER-016434), MAnAGER (POCI-01-0145-FEDER-028040), SUSpENsE (CENTRO-01-0145-FEDER-000006) and RTCARE (POCI-01-0145-FEDER-028030).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Abo-Sinna MA, Baky IA (2007) Interactive balance space approach for solving multi-level multi-objective programming problems. Inf Sci 177(16):3397–3410. CrossRefGoogle Scholar
  2. Alves MJ, Antunes CH (2018a) A differential evolution algorithm to semivectorial bilevel problems. In: Nicosia G et al (eds) Machine learning, optimization, and big data MOD 2017. Lecture notes in computer science. Springer, Cham, pp 172–185. CrossRefGoogle Scholar
  3. Alves MJ, Antunes CH (2018b) A semivectorial bilevel programming approach to optimize electricity dynamic time-of-use retail pricing. Comput Oper Res 92:130–144. CrossRefGoogle Scholar
  4. Alves MJ, Antunes CH, Carrasqueira P (2015) 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.
  5. Alves MJ, Antunes CH, Costa JP (2019) Multiobjective bilevel programming: concepts and perspectives of development. In: Doumpos M et al (eds) New perspectives in multiple criteria decision making: innovative applications and case studies. Springer, Cham, pp 267–293. CrossRefGoogle Scholar
  6. Ankhili Z, Mansouri A (2009) An exact penalty on bilevel programs with linear vector optimization lower level. Eur J Oper Res 197(1):36–41. CrossRefGoogle Scholar
  7. Bard J (1998) Nonconvex optimization and its applications. Kluwer Academic, Dordrecht. CrossRefGoogle Scholar
  8. Bonnel H (2006) Optimality conditions for the semivectorial bilevel optimization problem. Pac J Optim 2(3):447–468Google Scholar
  9. Calvete H, Galé C (2011) On linear bilevel problems with multiple objectives at the lower level. Omega 39(1):33–40. CrossRefGoogle Scholar
  10. Carrasqueira P, Alves MJ, Antunes CH (2015) A bi-level multiobjective PSO algorithm. In: Gaspar-Cunha A, Antunes CH, Coello Coello C (eds) Evolutionary multi-criterion optimization (EMO 2015). Lecture notes in computer science, vol 9018. Springer, Berlin, pp 263–276. CrossRefGoogle Scholar
  11. Colson B, Marcotte P, Savard G (2005) Bilevel programming: a survey. 4OR 3(2):87–107. CrossRefGoogle Scholar
  12. Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153(1):235–256. CrossRefGoogle Scholar
  13. Dassanayaka SM (2010) Methods of variational analysis in pessimistic bilevel programming. Wayne State University Dissertations, 126Google Scholar
  14. Deb K, Sinha A (2009) Solving bilevel multi-objective optimization problems using evolutionary algorithms. In: Proceeedings of EMO 2009, vol 5467. Lecture notes in computer science. Springer, Berlin, pp 110–124. CrossRefGoogle Scholar
  15. Deb K, Sinha A (2010) An efficient and accurate solution methodology for bilevel multi-objective programming problems using a hybrid evolutionary-local-search algorithm. Evol Comput 18(3):403–449. CrossRefGoogle Scholar
  16. Dempe S (2002) Foundations of bilevel programming. Springer, New York. CrossRefGoogle Scholar
  17. Eichfelder G (2010) Multiobjective bilevel optimization. Math Program 123(2):419–449. CrossRefGoogle Scholar
  18. Gupta A, Ong Y (2015) An evolutionary algorithm with adaptive scalarization for multiobjective bilevel programs. In: 2015 IEEE congress on evolutionary computation (CEC), Sendai, pp 1636–1642.
  19. Labbé M, Violin A (2013) Bilevel programming and price setting problems. 4OR 11(1):1–30. CrossRefGoogle Scholar
  20. Liu B et al (2014) Optimality conditions for pessimistic semivectorial bilevel programming problems. J Inequal Appl 2014:41. CrossRefGoogle Scholar
  21. Lucchetti R, Mignanego F, Pieri G (1987) Existence theorems of equilibrium points in stackelberg. Optimization 18(6):857–866. CrossRefGoogle Scholar
  22. Lv Y, Chen J (2016) A discretization iteration approach for solving a class of semivectorial bilevel programming problem. J Nonlinear Sci Appl 9(5):2888–2899. CrossRefGoogle Scholar
  23. Lv Y, Wan Z (2014) A solution method for the optimistic linear semivectorial bilevel optimization problem. J Inequal Appl 1:164. CrossRefGoogle Scholar
  24. Pieume C et al (2011) Solving bilevel linear multiobjective programming problems. Am J Oper Res 1:214–219. CrossRefGoogle Scholar
  25. Price K, Storn RM, Lampinen JA (2006) Differential evolution: a practical approach to global optimization. Springer, Berlin. CrossRefGoogle Scholar
  26. Ren A, Wang Y (2016) A novel penalty function method for semivectorial bilevel programming problem. Appl Math Model 40(1):135–149. CrossRefGoogle Scholar
  27. Ruuska S, Miettinen K (2012) Constructing evolutionary algorithms for bilevel multiobjective optimization. In: 2012 IEEE congress on evolutionary computation (CEC), IEEE, pp 1–7.
  28. Shi X, Xia H (1997) Interactive bilevel multi-objective decision making. J Oper Res Soc 48(9):943–949. CrossRefGoogle Scholar
  29. Shi X, Xia H (2001) Model and interactive algorithm of bi-level multi-objective decision-making with multiple interconnected decision makers. J Multi-Criteria Decis Anal 10:27–34. CrossRefGoogle Scholar
  30. Sinha A et al (2016) Solving bilevel multicriterion optimization problems with lower level decision uncertainty. IEEE Trans Evol Comput 20(2):199–217. CrossRefGoogle Scholar
  31. Sinha A, Malo P, Deb K (2017) Evolutionary bilevel optimization: an introduction and recent advances. In: Bechikh S, Datta R, Gupta A (eds) Recent advances in evolutionary multi-objective optimization. Springer, Cham, pp 71–103. CrossRefGoogle Scholar
  32. Sinha A, Malo P, Deb K (2018) A review on bilevel optimization: from classical to evolutionary approaches and applications. IEEE Trans Evol Comput 22(2):276–295. CrossRefGoogle Scholar
  33. Vicente L, Calamai P (1994) Bilevel and multilevel programming: a bibliography review. J Global Optim 5(3):291–306. CrossRefGoogle Scholar
  34. Zhang T et al (2013) Solving high dimensional bilevel multiobjective programming problem using a hybrid particle swarm optimization algorithm with crossover operator. Knowl-Based Syst 53:13–19. CrossRefGoogle Scholar
  35. Zheng Y, Wan Z (2011) A solution method for semivectorial bilevel programming problem via penalty method. J Appl Math Comput 37(1–2):207–219. CrossRefGoogle Scholar
  36. Zheng Y, Chen J, Cao X (2014) A global solution method for semivectorial bilevel programming problem. Filomat 28(8):1619–1627. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CeBER and Faculty of EconomicsUniversity of CoimbraCoimbraPortugal
  2. 2.DEECUniversity of Coimbra, Polo 2CoimbraPortugal
  3. 3.INESC CoimbraCoimbraPortugal

Personalised recommendations