The Constrained $L_p$-$L_q$ Basis Pursuit Denoising Problem

In this paper, we consider the constrained $L_p$-$L_q$ basis pursuit denoising problem, which aims to find a minimizer of $\|\bf{x}\|_p^p$ subject to $\|A\bf{x}-\bf{b}\|_q\leq\sigma$ for given $A \in \mathbb{R}^{m \times n}$, $\bf{b}\in\mathbb{R}^m$, $\sigma \geq0$, $0\leq p\leq1$ and $q \geq 1$. We first study the 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