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Exploiting Sparsity in Semidefinite Programming via Matrix Completion II: Implementation and Numerical Results

Kazuhide Nakata (nakata***at***zzz.t.u-tokyo.ac.jp)
Katsuki Fujisawa (fujisawa***at***is-mj.archi.kyoto-u.ac.jp)
Mituhiro Fukuda (mituhiro***at***is.titech.ac.jp)
Masakazu Kojima (kojima***at***is.titech.ac.jp)
Kazuo Murota (murota***at***kurims.kyoto-u.ac.jp)

Abstract: 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.

Keywords: Semidefinite programming, Primal-dual interior-point method, Matrix completion problem, Clique tree, Numerical results

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: Mathematical Programming Vol.95 303-327 (2003). Mathematical Programming Vol.95 303-327 (2003).

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Entry Submitted: 03/06/2001
Entry Accepted: 03/06/2001
Entry Last Modified: 04/30/2004

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