Combined approach with second-order optimality conditions for bilevel programming problems

TL;DR

This paper introduces a novel combined approach incorporating second-order optimality conditions into bilevel programming, enhancing the derivation of necessary optimality conditions when traditional methods fail.

Contribution

It extends the combined approach by integrating second-order conditions as constraints, improving the analysis of constraint qualifications and optimality in bilevel problems.

Findings

Adding second-order conditions facilitates necessary optimality conditions.

The approach improves the handling of constraint qualifications.

It offers advantages over first-order methods in complex problems.

Abstract

In this paper, we propose a combined approach with second-order optimality conditions of the lower level problem to study constraint qualifications and optimality conditions for bilevel programming problems. The new method is inspired by the combined approach developed by Ye and Zhu in 2010, where the authors combined the classical first-order and the value function approaches to derive new necessary optimality conditions. In our approach, we add a second-order optimality condition to the combined program as a new constraint. We show that when all known approaches fail, adding the second-order optimality condition as a constraint makes the corresponding partial calmness condition and the resulting necessary optimality condition easier to hold. We also give some discussions on advantages and disadvantages of the combined approaches with the first-order and the second-order information.