Lifted Stationary Points of Sparse Optimization with Complementarity Constraints

TL;DR

This paper introduces a new approach to find lifted stationary points in sparse optimization problems with complementarity constraints, using a continuous relaxation and an augmented Lagrangian method, demonstrating practical efficiency.

Contribution

It defines MPCC lifted-stationarity, proposes an approximation algorithm with convergence guarantees, and applies it to sparse problems with complementarity constraints.

Findings

The relaxation problem shares the same global minimizers as the original.

The proposed algorithm converges to an MPCC lifted-stationary point.

The method effectively finds sparse solutions in practical problems.

Abstract

We aim to compute lifted stationary points of a sparse optimization problem (P0) with complementarity constraints. We define a continuous relaxation problem (Rv) that has the same global minimizers and optimal value with problem (P0). Problem (Rv) is a mathematical program with complementarity constraints (MPCC) and a difference-of-convex (DC) objective function. We define MPCC lifted-stationarity of (Rv) and show that it is weaker than directional stationarity, but stronger than Clarke stationarity for local optimality. Moreover, we propose an approximation method to solve (Rv) and an augmented Lagrangian method to solve its subproblem, which relaxes the equality constraint in (Rv) with a tolerance. We prove the convergence of our algorithm to an MPCC lifted-stationary point of problem (Rv) and use a sparse optimization problem with vertical linear complementarity constraints to demonstrate the efficiency of our algorithm on finding sparse solutions in practice.