A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD.

Abstract

Semidefinite programming (SDP) is an important branch of optimization and has wide range of applications: engineering, industry, chemistry, mathematics, etc. However, obtaining very accurate optimum for a semidefinite programming is difficult in general, especially for ill-posed ones. In this paper, we evaluated numerically highly accurate SDP solvers; SDPA-GMP, -QD and -DD, which employ multiple-precision arithmetic and mainly developed by our group. We applied to some problems from SDPLIB benchmark which contains some ill-posed problems as well. The SDPA-GMP, and -QD solved problems very accurately, whereas SDPA failed or obtained inaccurate optima. The SDPA-DD may be used for compensation for accuracy and speed. We also investigated the convergence behaviors which agreed well what theories indicated.