Abstract

In recent years, interior point algorithms have been used successfully for solving medium to large-size linear programming (LP) problems. In this paper we describe a highly parallel formulation of the interior point algorithm. A key component of the interior point algorithm is the solution of a sparse system of linear equations using Cholesky factorization. The performance of parallel Cholesky factorization is determined by (a) the communication overhead incurred by the algorithm, and (b) the load imbalance among the processors. In our parallel interior point algorithm, we use our recently developed parallel multifrontal algorithm that has the smallest communication overhead over all parallel algorithms for Cholesky factorization developed to date. The computation imbalance depends on the shape of the elimination tree associated with the sparse system reordered for factorization. To balance the computation, we implemented and evaluated four di#erent ordering algorithms. Among these algorithms, Kernighan-Lin and spectral nested dissection yield the most balanced elimination trees and greatly increase the amount of parallelism that can be exploited. Our preliminary implementation achieves a speedup as high as 108 on 256-processor nCUBE 2 on moderate-size problems.

Citations