Second-order optimality conditions for optimization problems with generalized equation constraints

TL;DR

This paper establishes new second-order optimality conditions for complex optimization problems with generalized equation constraints, including MPVIs and bilevel programs.

Contribution

It develops comprehensive variational analysis for these constraints, providing novel second-order optimality conditions even for specific problem classes.

Findings

Derived second-order optimality conditions for MPVIs.

Developed variational analysis of intricate constraint systems.

Completed the study from a companion paper with new results.

Abstract

This paper provides second-order optimality conditions for optimization problems with generalized equation constraints (GEPs), a framework that encompasses several important and challenging models in mathematical programming, including mathematical programs with variational inequality constraints (MPVIs) and bilevel programs. The obtained optimality conditions are novel even for these particular problem classes. As an application, second-order optimality conditions for MPVIs are detailed. The technical key lies in developing first- and second-order variational analysis of the highly intricate constraint system, which is needed to capture the local curvature of the feasible set entering these optimality conditions. Part of this task was already carried out in our companion paper \cite{BeGfrYeZhangZhou}, and here we complete the study. Comprehensive variational analysis results are derived, which are of independent interest.