A parallel interior point decomposition algorithm forblock angular semidefinite programs

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 phase1, we suitably modify the matrix completion scheme of Fukuda et al. (SIAM J. Optim. 11:647-674, 2000) 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 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 (Message Passing Interface) implementation of the decomposition algorithm on the distributed Henry2 cluster with the OpenMP version of CSDP (Borchers and Young in Comput. Optim. Appl. 37:355-369, 2007) 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. (Optim. Methods Softw. 21:17-39, 2006), 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.