, Volume 143, Issue 1-2, pp 371-383
Date: 30 Nov 2012

Complexity of unconstrained \(L_2-L_p\) minimization

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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.