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