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.