# Sparse Solutions of a Class of Constrained Optimization Problems

## TL;DR

This paper investigates properties of solutions to a class of sparse optimization problems involving nonconvex and non-Lipschitz objectives, providing bounds, solution set characterizations, and an algorithm with convergence guarantees.

## Contribution

It offers new theoretical insights into the structure of solutions for sparse optimization with nonconvex penalties and proposes an effective smoothing penalty method for solving such problems.

## Key findings

- Optimal solutions are on the boundary of the feasible set when 0<p<1.
- The solution set for 0<p<1 is finite for q in {1,∞}.
- The proposed smoothing penalty method converges to a KKT point under mild conditions.

## Abstract

In this paper, we consider a well-known sparse optimization problem that aims to find a sparse solution of a possibly noisy underdetermined system of linear equations. Mathematically, it can be modeled in a unified manner by minimizing $\|\bf{x}\|_p^p$ subject to $\|A\bf{x}-\bf{b}\|_q\leq\sigma$ for given $A \in \mathbb{R}^{m \times n}$, $\bf{b}\in\mathbb{R}^m$, $\sigma \geq0$, $0\leq p\leq 1$ and $q \geq 1$. We then study various properties of the optimal solutions of this problem. Specifically, without any condition on the matrix $A$, we provide upper bounds in cardinality and infinity norm for the optimal solutions, and show that all optimal solutions must be on the boundary of the feasible set when $0<p<1$. Moreover, for $q \in \{1,\infty\}$, we show that the problem with $0<p<1$ has a finite number of optimal solutions and prove that there exists $0<p^*<1$ such that the solution set of the problem with any $0<p<p^*$ is contained in the solution set of the problem with $p=0$ and there further exists $0<\bar{p}<p^*$ such that the solution set of the problem with any $0<p\leq\bar{p}$ remains unchanged. An estimation of such $p^*$ is also provided. In addition, to solve the constrained nonconvex non-Lipschitz $L_p$-$L_1$ problem ($0<p<1$ and $q=1$), we propose a smoothing penalty method and show that, under some mild conditions, any cluster point of the sequence generated is a KKT point of our problem. Some numerical examples are given to implicitly illustrate the theoretical results and show the efficiency of the proposed algorithm for the constrained $L_p$-$L_1$ problem under different noises.