Decomposed block Cholesky factorization in the Karmarkar algorithm: Solving a class of super large LP problems

Abstract

With the introduction of the Karmarkar algorithm hope for optimally solving “super large” LP problems, often encountered in the communication industry, had risen high. However, a general code of the original Karmarkar algorithm takes a long computation time to solve such problems. Employing proper factorization schemes substantially reduces the computation time of the Karmarkar algorithm for solving super large linear programming (LP) problems. In this paper a modified version of the Karmarkar algorithm is described, and the complexities involved in applying the algorithm to a class of super large LP problems are identified. A factorization scheme, “decomposed block Cholesky factorization”, is proposed in order to reduce the complexities of a matrix inversion operation that occurs within every iteration of the Karmarkar algorithm. The factorization scheme capitalizes on the generally well-behaved block structure of the constraint matrix of the super large LP problems. The computation time required for solving a class of super large LP problems, using this scheme, is estimated to be reduced by a factor of two orders of magnitude. The proposed factorization scheme is even more promising in solving the super large LP problems on parallel processors using the Karmarkar algorithm.