First-Order Optimality Conditions for Mathematical Programming with Equilibrium Constraints

TL;DR

This paper develops a geometric framework for analyzing first-order optimality conditions in mathematical programs with equilibrium constraints, highlighting limitations of classical methods and providing new constraint qualifications.

Contribution

It introduces a first-principles approach focusing on the geometric structure of MPECs, clarifying stationarity and tangent cone concepts for better analysis.

Findings

Classical KKT-based approaches often require restrictive assumptions.

A detailed characterization of tangent cones at feasible points is provided.

The framework offers practical guidance for handling challenging MPECs.

Abstract

We present a systematic introduction to first-order optimality conditions for mathematical programs with equilibrium constraints (MPECs), emphasizing the limitations of classical nonlinear programming techniques. The goal is twofold. First, we explain why a direct application of standard optimality conditions -- based on reformulating MPECs via KKT systems or differentiable exact penalty functions -- is often inadequate, as such approaches typically require strong and restrictive assumptions, including nondegeneracy and smoothness conditions. Second, we develop a first-principles framework for analyzing MPECs by focusing on the geometric structure of the feasible region. In particular, we study stationarity concepts and provide a detailed characterization of the tangent cone at feasible points, which leads to appropriate constraint qualifications tailored to MPECs. These results form the foundation for rigorous first-order analysis and clarify the relationship between the original MPEC formulation and its KKT-based representation, offering practical guidance for handling these inherently challenging optimization problems.