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A Penalty Method for Rank Minimization Problems in Symmetric Matrices

Xin Shen(shenxinhust***at***gmail.com)
John E. Mitchell(mitchj***at***rpi.edu)

Abstract: The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal solution is a KKT point. We develop a penalty formulation of the problem. We present calmness results for locally optimal solutions to the penalty formulation. We also develop a proximal alternating linearized minimization (PALM) scheme for the penalty formulation, and investigate the incorporation of a momentum term into the algorithm. Computational results are presented.

Keywords: Rank minimization, penalty methods, alternating minimization

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 3: Complementarity and Variational Inequalities

Citation: Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA. January 2017

Download: [PDF]

Entry Submitted: 01/12/2017
Entry Accepted: 01/12/2017
Entry Last Modified: 01/12/2017

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