Numerical Stability in Linear Programming and Semidefinite Programming

Abstract

We study numerical stability for interior-point methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the difficulties inherent in current methods and present robust algorithms. <br /><br /> We start with the error bound analysis of the search directions for the normal equation approach for LP. Our error analysis explains the surprising fact that the ill-conditioning is not a significant problem for the normal equation system. We also explain why most of the popular LP solvers have a default stop tolerance of only 10^-8 when the machine precision on a 32-bit computer is approximately 10^-16. <br /><br /> We then propose a simple alternative approach for the normal equation based interior-point method. This approach has better numerical stability than the normal equation based method. Although, our approach is not competitive in terms of CPU time for the NETLIB problem set, we do obtain higher accuracy. In addition, we obtain significantly smaller CPU times compared to the normal equation based direct solver, when we solve well-conditioned, huge, and sparse problems by using our iterative based linear solver. Additional techniques discussed are: crossover; purification step; and no backtracking. <br /><br /> Finally, we present an algorithm to construct SDP problem instances with prescribed strict complementarity gaps. We then introduce two <em>measures of strict complementarity gaps</em>. We empirically show that: (i) these measures can be evaluated accurately; (ii) the size of the strict complementarity gaps correlate well with the number of iteration for the SDPT3 solver, as well as with the local asymptotic convergence rate; and (iii) large strict complementarity gaps, coupled with the failure of Slater's condition, correlate well with loss of accuracy in the solutions. In addition, the numerical tests show that there is no correlation between the strict complementarity gaps and the geometrical measure used in [31], or with Renegar's condition number.