A new problem qualification based on approximate KKT conditions for Lipschitzian optimization with application to bilevel programming

TL;DR

This paper introduces the Subset Mangasarian-Fromovitz Condition (subMFC), a new problem qualification based on approximate KKT conditions, which is weaker than existing qualifications and applicable to bilevel optimization.

Contribution

The paper proposes subMFC, a novel problem qualification derived from nonsmooth approximate KKT conditions, extending the theoretical framework for Lipschitzian optimization and bilevel programming.

Findings

subMFC is strictly weaker than quasinormality

subMFC can hold even if other qualifications fail

applies to bilevel optimization with lower-level value function reformulation

Abstract

When dealing with general Lipschitzian optimization problems, there are many problem classes where even weak constraint qualifications fail at local minimizers. In contrast to a constraint qualification, a problem qualification does not only rely on the constraints but also on the objective function to guarantee that a local minimizer is a Karush-Kuhn-Tucker (KKT) point. For example, calmness in the sense of Clarke is a problem qualification. In this article, we introduce the Subset Mangasarian-Fromovitz Condition (subMFC). This new problem qualification is derived by means of a nonsmooth version of the approximate KKT conditions, which hold at every local minimizer without further assumptions. A comparison with existing constraint and problem qualifications reveals that subMFC is strictly weaker than quasinormality and can hold even if the local error bound condition, the cone-continuity property, the Guignard constraint qualification and calmness are violated. Furthermore, we emphasize the power of the new problem qualification within the context of bilevel optimization. More precisely, under mild assumptions on the problem data, we suggest a version of subMFC that is tailored to the lower-level value function reformulation. It turns out that this new condition can be satisfied even if the widely used partial calmness condition does not hold.