Complexity of unconstrained L_2-L_p minimization

We consider the unconstrained $L_q$ - $L_p$ minimization: find a minimizer of $\Vert Ax-b\Vert ^q_q+\lambda \Vert x\Vert ^p_p$ for given $A \in R^{m\times n}$ , $b\in R^m$ and parameters $\lambda >0$ , $p\in 0, 1)$ and $q\ge 1$ . This problem has been studied extensively in many areas. Especially, for the case when $q=2$ , this problem is known as the $L_2-L_p$ minimization problem and has found its applications in variable selection problems and sparse least squares fitting for high dimensional data. Theoretical results show that the minimizers of the $L_q$ - $L_p$ problem have various attractive features due to the concavity and non-Lipschitzian property of the regularization function $\Vert \cdot \Vert ^p_p$ . In this paper, we show that the $L_q$ - $L_p$ minimization problem is strongly NP-hard for any $p\in 0,1)$ and $q\ge 1$ , including its smoothed version. On the other hand, we show that, by choosing parameters $(p,\lambda )$ carefully, a minimizer, global or local, will have certain desired sparsity. We believe that these results provide new theoretical insights to the studies and applications of the concave regularized optimization problems.