Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs

TL;DR

This paper introduces a generalized RCPLD constraint qualification that is weaker than traditional ones, extends it to complex systems, and applies it to bilevel programming problems, ensuring error bounds and optimality conditions.

Contribution

The paper extends RCPLD to general systems with diverse constraints and demonstrates its application to bilevel programs, providing new theoretical insights.

Findings

RCPLD is a valid constraint qualification for complex systems.

The generalized RCPLD ensures error bounds under certain regularity conditions.

Application to bilevel programs shows practical relevance.

Abstract

Relaxed constant positive linear dependence constraint qualification (RCPLD) for a system of smooth equalities and inequalities is a constraint qualification that is weaker than the usual constraint qualifications such as Mangasarian Fromovitz constraint qualification and the linear constraint qualification. Moreover RCPLD is known to induce an error bound property. In this paper we extend RCPLD to a very general feasibility system which may include Lipschitz continuous inequality constraints, complementarity constraints and abstract constraints. We show that this RCPLD for the general system is a constraint qualification for the optimality condition in terms of limiting subdifferential and limiting normal cone and it is a sufficient condition for the error bound property under the strict complementarity condition for the complementarity system and Clarke regularity conditions for the inequality constraints and the abstract constraint set. Moreover we introduce and study some sufficient conditions for RCPLD including the relaxed constant rank constraint qualification (RCRCQ). Finally we apply our results to the bilevel program.