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

Mituhiro Fukuda, Masakazu Kojima, Kazuo Murota and Kazuhide Nakata

A critical disadvantage of primal-dual interior-point methods against dual interior-point methods for large scale SDPs (semidefinite programs) has been that the primal positive semidefinite variable matrix becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidefinite 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 semidefinite variable matrix into an SDP having multiple but smaller size positive semidefinite variable matrices to which we can effectively 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 definite matrix completion, and report some numerical results.

Research Report B-358, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan. (Also issued as RIMS Preprint No. 1264, Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.) December 1999.

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