# MIP and Set Covering approaches for Sparse Approximation

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.
Document type :
Conference papers

https://hal.archives-ouvertes.fr/hal-03542681
Contributor : Matthieu Kowalski Connect in order to contact the contributor
Submitted on : Tuesday, January 25, 2022 - 3:00:31 PM
Last modification on : Wednesday, May 18, 2022 - 3:34:49 AM

### Identifiers

• HAL Id : hal-03542681, version 1
• ARXIV : 2009.06312

### Citation

Diego Delle Donne, Matthieu Kowalski, Leo Liberti. MIP and Set Covering approaches for Sparse Approximation. iTwist 2020 - International Traveling Workshop on Interactions between low-complexity data models and Sensing Techniques, Jun 2020, Nantes, France. ⟨hal-03542681⟩

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