A new proximity function generating the best known iteration bounds for both large-update and small-update interior-point methods

Abstract

AbstractInterior-Point Methods (IPMs) are not only very effective in practice for solving linear optimization problems but also have polynomial-time complexity. Despite the practical efficiency of large-update algorithms, from a theoretical point of view, these algorithms have a weaker iteration bound with respect to small-update algorithms. In fact, there is a significant gap between theory and practice for large-update algorithms. By introducing self-regular barrier functions, Peng, Roos and Terlaky improved this gap up to a factor of log n. However, checking these self-regular functions is not simple and proofs of theorems involving these functions are very complicated. Roos el al. by presenting a new class of barrier functions which are not necessarily self-regular, achieved very good results through some much simpler theorems. In this paper we introduce a new kernel function in this class which yields the best known complexity bound, both for large-update and small-update methods.