Complexity of unconstrained \(L_2L_p\) minimization
 Xiaojun Chen,
 Dongdong Ge,
 Zizhuo Wang,
 Yinyu Ye
 … show all 4 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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.
Inside
Within this Article
 Introduction
 Choosing the parameter \(\lambda \) for sparsity
 The \(L_2L_p\) problem is strongly NPhard
 Bounds \(\beta (k)\) and \(\gamma (k)\) for asymptotic properties
 References
 References
Other actions
 Chartrand, R.: Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process. Lett. 14, 707–710 (2007) CrossRef
 Chartrand, R., Staneva, V.: Restricted isometry properties and nonconvex compressive sensing. Inverse Probl. 24, 1–14 (2008) CrossRef
 Chen, X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of \(\ell _2\ell _p\) minimization. SIAM J. Sci. Comput. 32, 2832–2852 (2010) CrossRef
 Chen, X., Zhou, W.: Convergence of reweighted \(l_1\) minimization algorithms and unique solution of truncated \(l_p\) minimization. Department of Applied Mathematics, The Hong Kong Polytechnic University, Preprint (2010)
 Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Soc. 96, 1348–1360 (2001) CrossRef
 Foucart, S., Lai, M.J.: Sparsest solutions of underdetermined linear systems via \(l_q\) minimization for \(0<q\le 1\) . Appl. Comput. Harmon. Anal. 26, 395–407 (2009) CrossRef
 Frank, I.E., Freidman, J.H.: A statistical view of some chemometrics regression tools (with discussion). Technometrics 35, 109–148 (1993) CrossRef
 Garey, M.R., Johnson, D.S.: “Strong” NPCompleteness results: motivation, examples, and implications. J. Assoc. Comput. Mach. 25, 499–508 (1978) CrossRef
 Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NPCompleteness. W. H. Freeman, New York (1979)
 Ge, D., Jiang, X., Ye, Y.: A note on the complexity of \(L_p\) minimization. Math. Program. 129, 285–299 (2011) CrossRef
 Huang, J., Horowitz, J.L., Ma, S.: Asymptotic properties of bridge estimators in sparse highdimensional regression models. Ann. Stat. 36, 587–613 (2008) CrossRef
 Knight, K., Fu, W.J.: Asymptotics for lassotype estimators. Ann. Stat. 28, 1356–1378 (2000) CrossRef
 Lai, M., Wang, Y.: An unconstrained \(l_q\) minimization with \(0 < q< 1\) for sparse solution of underdetermined linear systems. SIAM J. Optim. 21, 82–101 (2011) CrossRef
 Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995) CrossRef
 Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)
 Tibshirani, R.: Regression shrinkage and selection via the Lasso. J R. Stat. Soc. B 58, 267–288 (1996)
 Vazirani, V.: Approximation Algorithms. Springer, Berlin (2003) CrossRef
 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