A penalty method for rank minimization problems in symmetric matrices

A penalty method for rank minimization problems in symmetric matrices 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. This is a continuous and nonconvex reformulation of the rank minimization problem. 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Optimization and Applications Springer Journals

A penalty method for rank minimization problems in symmetric matrices

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Optimization; Operations Research, Management Science; Operations Research/Decision Theory; Statistics, general; Convex and Discrete Geometry
ISSN
0926-6003
eISSN
1573-2894
D.O.I.
10.1007/s10589-018-0010-6
Publisher site
See Article on Publisher Site

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. This is a continuous and nonconvex reformulation of the rank minimization problem. 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.

Journal

Computational Optimization and ApplicationsSpringer Journals

Published: May 30, 2018

References

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