MIP and Set Covering approaches for Sparse Approximation

TL;DR

This paper introduces MIP and set covering formulations for sparse approximation, providing valid inequalities, reformulations, and algorithms, with computational experiments demonstrating their effectiveness.

Contribution

It develops new MIP and set covering models with valid inequalities for sparse approximation, along with algorithms and heuristics, advancing solution methods for this problem.

Findings

Valid inequalities effectively strengthen the models

Algorithms show promising practical performance

Set covering approach is competitive with MIP methods

Abstract

The Sparse Approximation problem asks to find a solution $x$ such that $||y - Hx|| < \alpha$, for a given norm $||\cdot||$, minimizing the size of the support $||x||_0 := \#\{j \ |\ x_j \neq 0 \}$. We present valid inequalities for Mixed Integer Programming (MIP) formulations for this problem and we show that these families are sufficient to describe the set of feasible supports. This leads to a reformulation of the problem as an Integer Programming (IP) model which in turn represents a Minimum Set Covering formulation, thus yielding many families of valid inequalities which may be used to strengthen the models up. We propose algorithms to solve sparse approximation problems including a branch \& cut for the MIP, a two-stages algorithm to tackle the set covering IP and a heuristic approach based on Local Branching type constraints. These methods are compared in a computational experimentation with the goal of testing their practical potential.