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 a...

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Bibliographic Details
Published inComputational optimization and applications Vol. 71; no. 2; pp. 353 - 380
Main Authors Shen, Xin, Mitchell, John E.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.11.2018
Springer Nature B.V
Subjects
Convex and Discrete Geometry
Management Science
Mathematics
Mathematics and Statistics
Matrix methods
Operations Research
Operations Research/Decision Theory
Optimization
Statistics
90C53
Penalty methods
Alternating minimization
90C33
Rank minimization
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Summary: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.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-018-0010-6