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Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion

Title
Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion
Authors
Kim S.Kojima M.Mevissen M.Yamashita M.
Ewha Authors
김선영
SCOPUS Author ID
김선영scopus
Issue Date
2011
Journal Title
Mathematical Programming
ISSN
0025-5610JCR Link
Citation
vol. 129, no. 1, pp. 33 - 68
Indexed
SCI; SCIE; SCOPUS WOS scopus
Abstract
A basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities. The sparsity, characterized with a chordal graph structure, can be detected in the variable matrix or in a linear or nonlinear matrix-inequality constraint of the problem. We classify the sparsity in two types, the domain-space sparsity (d-space sparsity) for the symmetric matrix variable in the objective and/or constraint functions of the problem, which is required to be positive semidefinite, and the range-space sparsity (r-space sparsity) for a linear or nonlinear matrix-inequality constraint of the problem. Four conversion methods are proposed in this framework: two for exploiting the d-space sparsity and the other two for exploiting the r-space sparsity. When applied to a polynomial semidefinite program (SDP), these conversion methods enhance the structured sparsity of the problem called the correlative sparsity. As a result, the resulting polynomial SDP can be solved more effectively by applying the sparse SDP relaxation. Preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems. © 2010 Springer and Mathematical Optimization Society.
DOI
10.1007/s10107-010-0402-6
Appears in Collections:
자연과학대학 > 수학전공 > Journal papers
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