We are concerned with a class of weak linear bilevel programs with nonunique lower level solutions. For such problems, we give via an exact penalty method an existence theorem of solutions. Then, we propose an algorithm.  2004 Elsevier Inc. All rights reserved.

In this paper, we consider a two-level optimization problem with nonunique lower-level solutions. We give sufficient conditions ensuring the existence of solutions. ᮊ

In this paper, which is an extension of [4], we first show the existence of solutions to a class of Min Sup problems with linked constraints, which satisfy a certain property. Then, we apply our result to a class of weak nonlinear bilevel problems. Furthermore, for such a class of bilevel problems, we give a relationship with appropriate d.c. problems concerning the existence of solutions.

International audienceIn this paper we give a conjugate duality approach for a strong bilevel programming problem $(S)$. The approach is based on the use of a regularization of problem $(S)$ and the so-called Fenchel--Lagrange duality. We first show that the regularized problem of $(S)$ admits solutions and any accumulation point of a sequence of regularized solutions solves $(S)$. Then, via this duality approach, we establish necessary and sufficient optimality conditions for the regularized problem. Finally, necessary and sufficient optimality conditions are given for the initial problem $(S)$. We note that such an approach which allows us to apply the Fenchel--Lagrange duality to the class of strong bilevel programming problems is new in the literature. An application to a two-level resource allocation problem is given

International audienceThe paper deals with a strong-weak nonlinear bilevel problem which generalizes the well-known weak and strong ones. In general, the study of the existence of solutions to such a problem is a difficult task. So that, for a strong-weak nonlinear bilevel prob- lem, we first give a regularization based on the use of strict ε-solutions of the lower level problem. Then, via this regularization and under sufficient conditions, we show that the problem admits at least one solution. The obtained result is an extension and an improve- ment of some recent results appeared recently in the literature for both weak nonlinear bilevel programming problems and linear finite dimensional case