Optimization Workshop Notes for Mathematical Programming with Equilibrium Constraints (MPECs): Second-Order Optimality Conditions

TL;DR

This workshop notes provide a rigorous overview of second-order optimality conditions for MPECs, highlighting three main viewpoints and emphasizing conceptual understanding of curvature and regularity.

Contribution

It introduces a comprehensive framework for second-order conditions in MPECs, clarifying the roles of multiplier, implicit, and piecewise programming approaches.

Findings

Three main viewpoints for second-order conditions are explained.

Emphasis on the role of curvature and regularity in optimality conditions.

Provides a conceptual understanding of second-order optimality in MPECs.

Abstract

In this workshop, we present a compact but rigorous introduction to second-order optimality conditions for mathematical programs with equilibrium constraints (MPECs). We start from the classical nonlinear programming template, then explain why it fails in the equilibrium-constrained setting, and develop the three main viewpoints used in the literature: (i) multiplier-based conditions, (ii) implicit-programming conditions based on the solution map of the lower-level equilibrium system, and (iii) piecewise-programming conditions obtained by decomposing complementarity structure into smooth pieces. The emphasis is on conceptual structure, critical cones, strong regularity, and the exact role of curvature terms.