Complexity Analysis of Primal-Dual Interior-Point Methods for Linear Optimization Based on a New Parametric Kernel Function with a Trigonometric Barrier Term

X Z Cai,G Q Wang,M El Ghami,Y J Yue

doi:10.1155/2014/710158

Abstract

We introduce a new parametric kernel function, which is a combination of the classic kernel function and a trigonometric barrier term, and present various properties of this new kernel function. A class of large- and small-update primal-dual interior-point methods for linear optimization based on this parametric kernel function is proposed. By utilizing the feature of the parametric kernel function, we derive the iteration bounds for large-update methods,O(n2/3log⁡(n/ε)), and small-update methods,O(nlog⁡(n/ε)). These results match the currently best known iteration bounds for large- and small-update methods based on the trigonometric kernel functions.

Highlights

In this paper, we consider the linear optimization (LO) problem in standard form min {cTx : Ax = b, x ≥ 0}, (P)where A ∈ Rm×n with rank(A) = m, b ∈ Rm, and c ∈ Rn
We introduce a new parametric kernel function, which is a combination of the classic kernel function and a trigonometric barrier term, and present various properties of this new kernel function
We develop some new properties of the parametric kernel function, as well as the corresponding barrier function

Summary

Introduction

While the so-called small-update IPMs enjoy the best known worstcase iteration bounds but their performance in computational practice is poor This gap was reduced by Peng et al [2] who introduced the so-called self-regular kernel functions and designed primal-dual IPMs based on self-regular proximities for LO. El Ghami et al [6] first introduced a trigonometric kernel function for primal-dual IPMs in LO They established the worst case iteration bounds for largeand small-update methods, namely, O(n3/4 log(n/ε)) and O(√n log(n/ε)), respectively. Peyghami and Hafshejani [8] established the better iteration bound O(√n(log(n)) log(n/ε)) for large-update methods based on a new kernel function consisting of a trigonometric function in its barrier term.

Framework of Kernel-Based IPMs for LO

New Parametric Kernel Function and Its Properties

Analysis and Complexity of the Algorithms

Conclusions and Remarks