Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework

  • Fukuda M
  • Kojima M
  • Murota K
 et al. 
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Abstract

A critical disadvantage of primal-dual interior-point methods against dual interiorpoint methods for large scale SDPs (semide nite programs) has been that the primal positive semide nite variable matrix becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semide nite matrix completion, this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage. Our method is used in two ways. One is a conversion of a sparse SDP having a large scale positive semide nite variable matrix into an SDP having multiple but smaller size positive semide nite variable matrices to which we can e ectively apply any interior-point method for SDPs employing a standard block-diagonal matrix data structure. The other way is an incorporation of our method into primal-dual interior-point methods which we can apply directly to a given SDP. In Part II of this article, we will investigate an implementation of such a primal-dual interior-point method based on positive de nite matrix completion, and report some numerical results.

Author-supplied keywords

  • 05c50
  • 90c22
  • 90c51
  • ams subject classifications
  • chordal graph
  • matrix completion
  • pii
  • primal-dual interior-point method
  • problem
  • s1052623400366218
  • semidefinite programming

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Authors

  • Mituhiro Fukuda

  • Masakazu Kojima

  • Kazuo Murota

  • Kazuhide Nakata

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