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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Ewha Women’s University, 11-1 Dahyun-dong, Sudaemoon-gu, Seoul, 120-750, Korea

    Sunyoung Kim

  2. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo, 152-8552, Japan

    Masakazu Kojima, Martin Mevissen & Makoto Yamashita

Authors
  1. Sunyoung Kim
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  2. Masakazu Kojima
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  3. Martin Mevissen
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  4. Makoto Yamashita
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Corresponding author

Correspondence to Sunyoung Kim.

Additional information

S. Kim’s research was supported by KRF 2008-531-C00013 and KRF 2009-007-1314. M. Kojima’s research was Grant-in-Aid for Scientific Research (B) 19310096. M. Mevissen’s research was supported by the Doctoral Scholarship of the German Academic Exchange Service (DAAD). M. Yamashita’s research was supported by Grant-in-Aid for Young Scientists (B) 18710141.

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Kim, S., Kojima, M., Mevissen, M. et al. Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion. Math. Program. 129, 33–68 (2011). https://doi.org/10.1007/s10107-010-0402-6

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  • DOI: https://doi.org/10.1007/s10107-010-0402-6

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