Pull the Weighted Center Towards the Solution of LP

Abstract

In the paper of Liao and Todd [3] two weighted centers are introduced and used to design algorithms for solving systems of linear inequalities. The linear programming problems can be solved via the weighted centers of a sequence of linear inequalities formed by letting the objective be an extra constraint and increasing the lower bound corresponding to the objective function as long as it is possible. In this paper we study the second kind of weighted center of [3] which ismore computationally oriented and show that, under a regularity assumption, the weighted center of the linear inequality with the objective as an extra constraint converges to the solution of the linear programming problem under consideration as the upper bound corresponding to the objective function is pulled towards the infinity. We propose a relaxed version of one of the algorithms of [3]. This modified version does not try to find an accurate center during each iteration; instead, an approximate center which is the k-th feasible iterate is determined in the k-th iteration. We show that this modified algorithm finds an e-solution in finity many iterations. Some limited numerical results are presented to compare our algorithm with the simplex method and indicate that our algorithm is promising.