Matrix partitioning methods for interior point algorithms

Abstract

We consider here a linear programming problem whose rows of the constraint matrix can be partitioned into two parts. Such natural partitions exist in several special linear programs, including the assignment problem, the transportation problem, the generalized upper-bounded variable problem, the block diagonal linear program; and can also be used to induce sparsity patterns in Cholesky factorizations. In this paper, we propose a matrix partitioning method for interior point algorithms. The proposed method independently generates Cholesky factorizations of each part, and reduces the complexity to that of solving generally, a dense linear system involving several rank one updates of the identity matrix. Here, we propose solving this linear system by an inductive use of the Sherman-Morrison-Woodbury formula. The proposed method is easily implemented on a vector, parallel machine as well as on a distributed system. Such partitioning methods have been popular in the context of the simplex method, where the special structure of the basis matrix is exploited.