Large Step Path-Following Methods for Linear Programming, Part II: Potential Reduction Method

Abstract

The algorithm proposed in this paper is the affine polynomial potential reduction method, with new procedures for updating the lower bounds for an optimal solution of the linear programming problem. A method is developed for updating the lower bounds by large steps, with strict control over the duality gaps associated with each iterate. Two algorithms are obtained by this approach: the first one has complexity $O(nL)$ iterations, and a very simple updating procedure; the second one updates the lower bounds at points very near the central trajectory and achieves a complexity of $O(\sqrt{n} L)$ iterations.