An Implementable Proximal Point Algorithmic Framework for Nuclear Norm Minimization

Yong-Jin Liu(smalynus.edu.sg)
Defeng Sun(matsundfnus.edu.sg)
Kim-Chuan Toh(mattohkcnus.edu.sg)

Abstract: The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primal-dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner sub-problems are solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state-of-the-art algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature.

Keywords: Nuclear norm minimization, proximal point method, rank minimization, gradient projection method, accelerated proximal gradient method.

Category 1: Convex and Nonsmooth Optimization

Category 2: Linear, Cone and Semidefinite Programming

Citation: National University of Singapore, July, 2009.

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Entry Submitted: 07/13/2009
Entry Accepted: 07/13/2009
Entry Last Modified: 07/13/2009

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