Mathematical Programming

, Volume 143, Issue 1, pp 371–383

Complexity of unconstrained \(L_2-L_p\) minimization


    • Department of Applied MathematicsThe Hong Kong Polytechnic University
  • Dongdong Ge
    • Antai School of Economics and ManagementShanghai Jiao Tong University
  • Zizhuo Wang
    • Department of Industrial and System EngineeringUniversity of Minnesota
  • Yinyu Ye
    • Department of Management Science and EngineeringStanford University
Short Communication Series A

DOI: 10.1007/s10107-012-0613-0

Cite this article as:
Chen, X., Ge, D., Wang, Z. et al. Math. Program. (2014) 143: 371. doi:10.1007/s10107-012-0613-0


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 optimizationNonconvex optimizationVariable selectionSparse solution reconstructionBridge estimator

Mathematics Subject Classification (2010)


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© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2012