Optimization Workshop Notes for Mathematical Programming with Equilibrium Constraints Algorithms: Penalty Interior-Point, Implicit-Programming, and Piecewise SQP

TL;DR

This paper reviews various algorithms for solving mathematical programs with equilibrium constraints, emphasizing their models, convergence, and the distinctions between ideas and proven results.

Contribution

It unifies different algorithmic approaches for MPECs, clarifying their mechanisms and convergence assumptions, and discusses their theoretical and practical aspects.

Findings

PIPA is a classical approach for MPECs.

A monotone-LCP variant of PIPA explicitly controls complementarity decay.

Piecewise SQP applies SQP on locally smooth pieces for MPECs.

Abstract

In this workshop, we discuss several algorithms for mathematical programs with equilibrium constraints (MPECs). The unifying theme is that MPECs are optimization problems whose feasible set contains a lower-level equilibrium system, often written through complementarity or variational-inequality conditions. This destroys the smooth manifold or convex structure that standard nonlinear programming methods rely on. We focus on four algorithmic viewpoints: (i) the classical penalty interior-point algorithm (PIPA); (ii) a monotone-linear complementarity problem (LCP) variant of PIPA that explicitly controls complementarity decay; (iii) an implicit-programming descent method for variational inequality (VI)-constrained MPECs; (iv) piecewise SQP (PSQP), which applies SQP on locally selected smooth pieces. For each method we explain the model, the search direction subproblem, the globalization mechanism, and the meaning of the convergence result. Particular emphasis is placed on what the assumptions are really doing and on the distinction between an attractive algorithmic idea and a fully valid convergence theorem.