Complexity of unconstrained \(L_2L_p\) minimization
 Xiaojun Chen,
 Dongdong Ge,
 Zizhuo Wang,
 Yinyu Ye
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We consider the unconstrained \(L_q\)  \(L_p\) minimization: find a minimizer of \(\Vert Axb\Vert ^q_q+\lambda \Vert x\Vert ^p_p\) 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 \(L_2L_p\) 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 nonLipschitzian 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 NPhard 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.
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 Title
 Complexity of unconstrained \(L_2L_p\) minimization
 Journal

Mathematical Programming
Volume 143, Issue 12 , pp 371383
 Cover Date
 20140201
 DOI
 10.1007/s1010701206130
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Nonsmooth optimization
 Nonconvex optimization
 Variable selection
 Sparse solution reconstruction
 Bridge estimator
 90C26
 90C51
 Industry Sectors
 Authors

 Xiaojun Chen ^{(1)}
 Dongdong Ge ^{(2)}
 Zizhuo Wang ^{(3)}
 Yinyu Ye ^{(4)}
 Author Affiliations

 1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China
 2. Antai School of Economics and Management, Shanghai Jiao Tong University, Shanghai, China
 3. Department of Industrial and System Engineering, University of Minnesota, Minneapolis, MN, 55455, USA
 4. Department of Management Science and Engineering, Stanford University, Stanford, CA, 943054121, USA