A Lasry-Lions envelope approach for mathematical programs with complementarity constraints

TL;DR

This paper introduces a homotopy method using a Lasry-Lions double envelope to efficiently solve mathematical programs with complementarity constraints, with proven convergence and promising numerical results.

Contribution

It develops a novel smooth approximation technique for CCs and a homotopy algorithm with convergence guarantees, advancing solution methods for CC problems.

Findings

Converges to Mordukhovich and Clarke stationary points.

Provides worst-case complexity analysis.

Numerical results show effectiveness on benchmark problems.

Abstract

We propose a homotopy method for solving mathematical programs with complementarity constraints (CCs). The indicator function of the CCs is relaxed by a Lasry-Lions double envelope, an extension of the Moreau envelope that enjoys an additional smoothness property that makes it amenable to fast optimization algorithms. The proposed algorithm mimics the behavior of homotopy methods for systems of nonlinear equations or penalty methods for constrained optimization: it solves a sequence of smooth subproblems that progressively approximate the original problem, using the solution of each subproblem as the starting point for the next one. In the limiting setting, we establish the convergence to Mordukhovich and Clarke stationary points. We also provide a worst-case complexity analysis for computing an approximate stationary point. Preliminary numerical results on a suite of benchmark problems demonstrate the effectiveness of the proposed approach.