Is Pessimistic Bilevel Programming a Special Case of a Mathematical Program with Complementarity Constraints?

Abstract

One of the most commonly used methods for solving bilevel programming problems (whose lower level problem is convex) starts with reformulating it as a mathematical program with complementarity constraints. This is done by replacing the lower level problem by its Karush-Kuhn-Tucker optimality conditions. The obtained mathematical program with complementarity constraints is (locally) solved, but the question of whether a solution of the reformulation yields a solution of the initial bilevel problem naturally arises. The question was first formulated and answered negatively, in a recent work of Dempe and Dutta, for the so-called optimistic approach. We study this question for the pessimistic approach also in the case of a convex lower level problem with a similar answer. Some new notions of local solutions are defined for these minimaxtype problems, for which the relations are shown. Some simple counterexamples are given.

On implicit variables in optimization theory

“…Particularly, there may exist local minimizers of (Q) which do not correspond to local minimizers of (P). Similar issues are pointed out in the context of bilevel programming, see Aussel and Svensson (2019); Dempe and Dutta (2012); Dempe et al (2018); Dempe and Mehlitz (2020), w.r.t. the use of slack variables in logical programming, see Mehlitz (2020a,b), or cardinality-constrained optimization, see Burdakov et al (2016), and will be generalized in this paper.…”

“…local minimizers, see e.g. Aussel and Svensson (2019); Dempe and Dutta (2012); Dempe et al (2018). The authors in Adam et al (2018), where the particular setting C := R s − is discussed, focused on a qualitative comparison of the reformulation…”

Random multifunctions as the set minimizers of infinitely many differentiable random functions

“…The first example relies on the setting of Mathematical programming with equilibrium constraints. This class of problems has captured the attention of several researchers due to its intrinsic relation with Nash equilibrium and bilevel problems (see, e.g, [2,3,10]). The second example corresponds to the setting of two-stage stochastic programming.…”

Optimal design of exchange networks with blind inputs and its application to Eco-industrial parks

“…Then they implemented a Branch-and-Bound heuristic to obtain an approximated optimal exchange network, solving at each iteration the problem described above. However, it is known that the MPCC problems, which is a particular class of mathematical programming with equilibrium constraints (MPEC), are hard to solve (see, e.g., Baumrucker et al, 2008;Tseveendorj, 2013;Luo et al, 1996 ) and the heuristic itself doesn't guarantee a real solution of the problem ( Aussel and Svensson, 2019;Dempe and Dutta, 2012 ). The literature on theoretical and algorithmic aspects of MPCC and MPEC problems is large and still an active field of research in mathematics.…”