Short Communication Series A

Mathematical Programming

, Volume 143, Issue 1, pp 371-383

First online:

Complexity of unconstrained \(L_2-L_p\) minimization

  • Xiaojun ChenAffiliated withDepartment of Applied Mathematics, The Hong Kong Polytechnic University Email author 
  • , Dongdong GeAffiliated withAntai School of Economics and Management, Shanghai Jiao Tong University
  • , Zizhuo WangAffiliated withDepartment of Industrial and System Engineering, University of Minnesota
  • , Yinyu YeAffiliated withDepartment of Management Science and Engineering, Stanford University

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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.


Nonsmooth optimization Nonconvex optimization Variable selection Sparse solution reconstruction Bridge estimator

Mathematics Subject Classification (2010)

90C26 90C51