A parallel primal–dual interior-point method for semidefinite programs using positive definite matrix completion

Abstract

A parallel computational method SDPARA-C is presented for SDPs (semidefinite programs). It combines two methods SDPARA and SDPA-C proposed by the authors who developed a software package SDPA. SDPARA is a parallel implementation of SDPA and it features parallel computation of the elements of the Schur complement equation system and a parallel Cholesky factorization of its coefficient matrix. SDPARA can effectively solve SDPs with a large number of equality constraints; however, it does not solve SDPs with a large scale matrix variable with similar effectiveness. SDPA-C is a primal–dual interior-point method using the positive definite matrix completion technique by Fukuda et al., and it performs effectively with SDPs with a large scale matrix variable, but not with a large number of equality constraints. SDPARA-C benefits from the strong performance of each of the two methods. Furthermore, SDPARA-C is designed to attain a high scalability by considering most of the expensive computations involved in the primal–dual interior-point method. Numerical experiments with the three parallel software packages SDPARA-C, SDPARA and PDSDP by Benson show that SDPARA-C efficiently solves SDPs with a large scale matrix variable as well as a large number of equality constraints with a small amount of memory.