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

  • Authors:

  • Kazuhide Nakata;Makoto Yamashita;Katsuki Fujisawa;Masakazu Kojima

  • Affiliations:

  • Department of Industrial Engineering and Management, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan;Department of Industrial Engineering and Management, Kanagawa University, 3-27-1, Rokkakubashi, Kanagawa-Ku, Yokohama, Kanagawa 221-8686, Japan;Department of Mathematical Sciences, Tokyo Denki University, Ishizuka, Hatoyama, Saitama 350-0394, Japan;Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan

  • Venue:
  • Parallel Computing
  • Year:
  • 2006

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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.