TY - JOUR

T1 - Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion

AU - Kim, Sunyoung

AU - Kojima, Masakazu

AU - Mevissen, Martin

AU - Yamashita, Makoto

PY - 2011/9

Y1 - 2011/9

N2 - 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.

AB - 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.

KW - Chordal Graph

KW - Matrix Inequalities

KW - Polynomial Optimization

KW - Positive Semidefinite Matrix Completion

KW - Semidefinite Program

KW - Sparsity

UR - http://www.scopus.com/inward/record.url?scp=80052599881&partnerID=8YFLogxK

U2 - 10.1007/s10107-010-0402-6

DO - 10.1007/s10107-010-0402-6

M3 - Article

AN - SCOPUS:80052599881

VL - 129

SP - 33

EP - 68

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1

ER -