An Extended Fenchel--Lagrange Duality Approach and Optimality Conditions for Strong Bilevel Programming Problems
Abdelmalek Aboussoror, Samir Adly, Fatima Ezzahra Saissi · SIAM Journal on Optimization · 2017
In 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.