Infeasible Interior-Point Methods for Linear Optimization Based on Large Neighborhood

Alireza Asadi,Cornelis Roos

doi:10.1007/s10957-015-0826-5

Abstract

In this paper, we design a class of infeasible interior-point methods for linear optimization based on large neighborhood. The algorithm is inspired by a full-Newton step infeasible algorithm with a linear convergence rate in problem dimension that was recently proposed by the second author. Unfortunately, despite its good numerical behavior, the theoretical convergence rate of our algorithm is worse up to square root of problem dimension.

Highlights

We consider the linear optimization (LO) problem in standard format, given by (P), and its dual problem given by (D)
Interior-point methods (IPMs) for LO are divided into two classes: feasible interiorpoint methods (FIPMs) and infeasible interior-point methods (IIPMs)
Motivated by the theoretical result which says that the kernel function ψq gives rise to the best-known theoretical iteration bound for large-update IIPMs based on kernel functions, we compare the performance of the algorithm described in the previous subsection based on both the logarithmic barrier function and the ψq -based barrier function

Summary

Introduction

We consider the linear optimization (LO) problem in standard format, given by (P), and its dual problem given by (D) (see Sect. 3). We consider the linear optimization (LO) problem in standard format, given by (P), and its dual problem given by (D) We assume that the problems (P) and (D) are feasible. This implies that both problems have optimal solutions and the same optimal value. Interior-point methods (IPMs) for LO are divided into two classes: feasible interiorpoint methods (FIPMs) and infeasible interior-point methods (IIPMs). FIPMs assume that a primal-dual strictly feasible point is available from which the algorithm can immediately start. In order to get such a solution several initialization methods have been presented, e.g., by Megiddo [1] and Anstreicher [2]. A disadvantage of the so-called combined phase-I and phase-II method of Anstreicher is that the feasible region must be nonempty and a lower bound on the optimal objective value must be known

Objectives

Results

Conclusion