Open Access
April 2016 Global solutions to folded concave penalized nonconvex learning
Hongcheng Liu, Tao Yao, Runze Li
Ann. Statist. 44(2): 629-659 (April 2016). DOI: 10.1214/15-AOS1380


This paper is concerned with solving nonconvex learning problems with folded concave penalty. Despite that their global solutions entail desirable statistical properties, they lack optimization techniques that guarantee global optimality in a general setting. In this paper, we show that a class of nonconvex learning problems are equivalent to general quadratic programs. This equivalence facilitates us in developing mixed integer linear programming reformulations, which admit finite algorithms that find a provably global optimal solution. We refer to this reformulation-based technique as the mixed integer programming-based global optimization (MIPGO). To our knowledge, this is the first global optimization scheme with a theoretical guarantee for folded concave penalized nonconvex learning with the SCAD penalty [J. Amer. Statist. Assoc. 96 (2001) 1348–1360] and the MCP penalty [Ann. Statist. 38 (2001) 894–942]. Numerical results indicate a significant outperformance of MIPGO over the state-of-the-art solution scheme, local linear approximation and other alternative solution techniques in literature in terms of solution quality.


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Hongcheng Liu. Tao Yao. Runze Li. "Global solutions to folded concave penalized nonconvex learning." Ann. Statist. 44 (2) 629 - 659, April 2016.


Received: 1 July 2014; Revised: 1 June 2015; Published: April 2016
First available in Project Euclid: 17 March 2016

zbMATH: 1337.62163
MathSciNet: MR3476612
Digital Object Identifier: 10.1214/15-AOS1380

Primary: 62J05
Secondary: 62J07

Keywords: Folded concave penalties , global optimization , high-dimensional statistical learning , MCP , nonconvex quadratic programming , SCAD , sparse recovery

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 2 • April 2016
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