Mathematical Programming

, Volume 143, Issue 1–2, pp 371–383

Complexity of unconstrained \(L_2-L_p\) minimization

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 optimization Nonconvex optimization Variable selection Sparse solution reconstruction Bridge estimator 

Mathematics Subject Classification (2010)

90C26 90C51 

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2012

Authors and Affiliations

  • Xiaojun Chen
    • 1
  • Dongdong Ge
    • 2
  • Zizhuo Wang
    • 3
  • Yinyu Ye
    • 4
  1. 1.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHong KongChina
  2. 2.Antai School of Economics and ManagementShanghai Jiao Tong UniversityShanghaiChina
  3. 3.Department of Industrial and System EngineeringUniversity of MinnesotaMinneapolisUSA
  4. 4.Department of Management Science and EngineeringStanford UniversityStanfordUSA

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