We present a two phase interior point decomposition framework for solving semidefinite (SDP) relaxations of sparse maxcut, stable set, and box constrained quadratic programs. In phase 1, we suitably modify the {\em matrix completion} scheme of Fukuda et al. \cite{fukuda_et_al} to preprocess an existing SDP into an equivalent SDP in the block-angular form. In phase 2, we solve the resulting block-angular SDP using a {\em regularized interior point decomposition} algorithm, in an iterative fashion between a master problem (a quadratic program); and decomposed and distributed subproblems (smaller SDPs) in a parallel and distributed high performance computing environment. We compare our MPI implementation of the decomposition algorithm on the distributed {\em Henry2} cluster with the OpenMP version of CSDP \cite{borchers_young} on the IBM Power5 shared memory system at NC State University. Our computational results indicate that the decomposition algorithm a) solves large SDPs to 2-3 digits of accuracy where CSDP runs out of memory; b) returns competitive solution times with the OpenMP version of CSDP, and c) attains a good parallel scalability. Comparing our results with Fujisawa et al. \cite{fujisawa_et_al}, we also show that a suitable modification of the matrix completion scheme can be used in the solution of larger SDPs than was previously possible.
Citation
Technical Report, Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, April 2008. Accepted for publication in "Computational Optimization and Applications". Also available at http://www4.ncsu.edu/~kksivara/publications/parallel-conic-blockangular.pdf
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View A PARALLEL interior point decomposition algorithm for block-angular semidefinite programs