Solving Mathematical Programs with Equilibrium Constraints as Nonlinear Programming: A New Framework

TL;DR

This paper introduces a novel framework for solving mathematical programs with equilibrium constraints by reformulating them as nonlinear programs and employing a trust-funnel strategy to ensure convergence to stationary points.

Contribution

It proposes a new algorithmic approach that combines unconstrained optimization of a complementarity measure with nonlinear programming techniques, addressing MPECs without requiring MFCQ.

Findings

The algorithm converges to an S-stationary point under MPEC-MFCQ.

The framework effectively balances feasibility and optimality improvements.

It generalizes trust region methods for MPECs.

Abstract

We present a new framework for the solution of mathematical programs with equilibrium constraints (MPECs). In this algorithmic framework, an MPECs is viewed as a concentration of an unconstrained optimization which minimizes the complementarity measure and a nonlinear programming with general constraints. A strategy generalizing ideas of Byrd-Omojokun's trust region method is used to compute steps. By penalizing the tangential constraints into the objective function, we circumvent the problem of not satisfying MFCQ. A trust-funnel-like strategy is used to balance the improvements on feasibility and optimality. We show that, under MPEC-MFCQ, if the algorithm does not terminate in finite steps, then at least one accumulation point of the iterates sequence is an S-stationary point.