A Primal-Dual Variant of the IRI-IMAI Algorithm for Linear Programming

Abstract

A local acceleration method for primal-dual potential-reduction algorithms is introduced. The method developed here uses modified Newton search directions to minimize the Tanabe-Todd-Ye (TTY) potential function, and can be regarded as a primal-dual variant of the Iri-Imai algorithm based on the multiplicative analogue of Karmarkar's potential function. When iterates are close to an optimal solution, the TTY potential function has negative curvature along the generated search directions. Therefore, large reductions in the potential function can be obtained, guaranteeing polynomial and quadratic convergence to nondegenerate solutions.

Quadratic convergence of potential-reduction methods for degenerate problems

“…This is a less precise analogue of Lemma 4.3 in our previous paper [12], but it does not require a nondegeneracy assumption. This is a less precise analogue of Lemma 4.3 in our previous paper [12], but it does not require a nondegeneracy assumption.…”

“…In our previous work [12], we established that such directions can be obtained for nondegenerate problems using a primal-dual variant of Iri and Imai's strategy. In our previous work [12], we established that such directions can be obtained for nondegenerate problems using a primal-dual variant of Iri and Imai's strategy.…”

“…In [12], such an analysis was performed with the assumption that the LP and its dual have nondegenerate solutions. 2 requires an accurate estimation of the different vectors and scalars appearing in equations (29) and (30).…”

“…The first one of these papers is by the present author [12], where a potential-reduction algorithm with polynomial and quadratic convergence for nondegenerate linear programming problems is developed. The first one of these papers is by the present author [12], where a potential-reduction algorithm with polynomial and quadratic convergence for nondegenerate linear programming problems is developed.…”

“…2, we review potential-reduction algorithms and summarize some of the results from [12] that will be useful in the remainder of this article. 2, we review potential-reduction algorithms and summarize some of the results from [12] that will be useful in the remainder of this article.…”