TY - JOUR

T1 - Exploiting sparsity in semidefinite programming via matrix completion II

T2 - Implementation and numerical results

AU - Nakata, Kazuhide

AU - Fujisawa, Katsuki

AU - Fukuda, Mituhiro

AU - Kojima, Masakazu

AU - Murota, Kazuo

PY - 2003/2/1

Y1 - 2003/2/1

N2 - In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two different ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primal-dual interior-point method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over different classes of SDPs show that these methods can be very efficient for some problems.

AB - In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two different ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primal-dual interior-point method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over different classes of SDPs show that these methods can be very efficient for some problems.

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U2 - 10.1007/s10107-002-0351-9

DO - 10.1007/s10107-002-0351-9

M3 - Article

AN - SCOPUS:1542337154

VL - 95

SP - 303

EP - 327

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 2

ER -