Sparse Solutions of a Class of Constrained Optimization Problems

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.