Introduction to Exact Penalization for Mathematical Programming with Equilibrium Constraints

TL;DR

This paper introduces exact penalty methods for nonlinear and equilibrium-constrained optimization, connecting classical optimality conditions with modern error bound theory and extending applicability through subanalytic geometry.

Contribution

It extends the theory of exact penalization beyond classical regularity conditions using subanalytic geometry and Lojasiewicz inequalities, with practical reformulations for MPECs.

Findings

Connection between optimality conditions and error bounds clarified.

Extension of exact penalization to broader analytic conditions.

Practical formulations for MPECs using residual-based penalties.

Abstract

We present a focused introduction to exact penalty methods for nonlinear programs and mathematical programs with equilibrium constraints (MPECs), emphasizing their connection to modern error bound theory. The goal is twofold. First, we explain how classical optimality conditions can be interpreted through exact penalization, and why such results typically rely on constraint regularity conditions that can be understood as error bounds on perturbations of feasible sets. We then highlight how recent developments based on subanalytic geometry and Lojasiewicz-type inequalities extend this framework beyond classical regularity assumptions, enabling exact penalization under broader analytic conditions. Second, we demonstrate how this theory can be applied in practice to MPECs by reformulating them via KKT systems and constructing exact penalty functions based on residual mappings. Particular attention is given to fractional-order penalties arising from Lojasiewicz error bounds, as well as to improved formulations for special problem classes where sharper exponents can be obtained. These developments provide both theoretical insight and practical guidance for analyzing and solving challenging constrained optimization problems.