Strong-weak Stackelberg Problems in Finite Dimensional Spaces

We are concerned with two-level optimization problems called strongweak Stackelberg problems, generalizing the class of Stackelberg problems in the strong and weak sense. In order to handle the fact that the considered two-level optimization problems may fail to have a solution under mild assumptions, we consider a regularization involving ε-approximate optimal solutions in the lower level problems. We prove the existence of optimal solutions for such regularized problems and present some approximation results when the parameter ǫ goes to zero. Finally, as an example, we consider an optimization problem associated to a best bound given in [2] for a system of nondifferentiable convex inequalities

A Fenchel-Lagrange Duality Approach for a Bilevel Programming Problem with Extremal-Value Function

International audienceIn this paper, for a bilevel programming problem (S) with an extremal-value function, we first give its Fenchel-Lagrange dual problem. Under appropriate assumptions, we show that a strong duality holds between them. Then, we provide optimality conditions for (S) and its dual. Finally, we show that the resolution of the dual problem is equivalent to the resolution of a one-level convex minimization problem

Weak Nonlinear Bilevel Problems : Existence of Solutions via Reverse Convex and Convex Maximization Problems

International audienceIn this paper, for a class of weak bilevel programming problems we provide sufficient conditions guaranteeing the existence of global solutions. These conditions are based on the use of reverse convex and convex maximization problems

Weak linear bilevel programming problems: existence of solutions via a penalty method

AbstractWe 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

New necessary and sufficient optimality conditions for strong bilevel programming problems

International audienceIn this paper we are interested in a strong bilevel programming problem (S). For such a problem, we establish necessary and sufficient global optimality conditions. Our investigation is based on the use of a regularization of problem (S) and some well-known global optimization tools. These optimality conditions are new in the literature and are expressed in terms of max–min conditions with linked constraints

Existence of solutions to weak nonlinear bilevel problems

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

Generalized Semi-Infinite Programming: Optimality Conditions Involving Reverse Convex Problems

International audienceThis article deals with a generalized semi-infinite programming problem (S). Under appropriate assumptions, for such a problem we give necessary and sufficient optimality conditions via reverse convex problems. In particular, a necessary and sufficient optimality condition reduces the problem (S) to a min-max problem constrained with compact convex linked constraints

BILEVEL PROGRAMS WITH EXTREMAL VALUE FUNCTION: GLOBAL OPTIMALITY

For a bilevel program with extremal value function, a necessary and sufficient condition for global optimality is given, which reduces the bilevel program to a max-min problem with linked constraints. Also, for the case where the extremal value function is polyhedral, this optimality condition gives the possibility of a resolution via a maximization problem of a polyhedral convex function over a convex set. Finally, this case is completed by an algorithm. 1