New complexity analysis of interior-point methods for the Cartesian P ∗ ( κ ) -SCLCP

Guoqiang Wang,Minmin Li,Yujing Yue,Xinzhong Cai

doi:10.1186/1029-242x-2013-285

Abstract

In this paper, we give a unified analysis for both large- and small-update interior-point methods for the Cartesian P ∗ (κ)-linear complementarity problem over symmetric cones based on a finite barrier. The proposed finite barrier is used both for determining the search directions and for measuring the distance between the given iterate and the μ-center for the algorithm. The symmetry of the resulting search directions is forced by using the Nesterov-Todd scaling scheme. By means of Euclidean Jordan algebras, together with the feature of the finite kernel function, we derive the iteration bounds that match the currently best known iteration bounds for large- and small-update methods. Furthermore, our algorithm and its polynomial iteration complexity analysis provide a unified treatment for a class of primal-dual interior-point methods and their complexity analysis.MSC:90C33, 90C51.

Highlights

Let (V, ◦) be an n-dimensional Euclidean Jordan algebra with rank r equipped with the standard inner product x, s = tr(x ◦ s)
We have described the scheme that defines the classical NT-direction for the Cartesian P∗(κ)-SCLCP
We have found a ε-approximate solution of the Cartesian P∗ (κ )-SCLCP

Summary

Introduction

This completes the proof of the lemma From this point on, the analysis of the algorithm follows almost completely the similar analyses in [ , ] with straightforward modifications that take into account the Cartesian P∗(κ)-property. 5.3 Decrease of the value of (v) during an inner iteration In what follows, we will show that the barrier function (v) in each inner iteration with the default step size α , as defined by ( ), is decreasing. The following theorem states the results which show that the default step size ( ) yields sufficient decrease of the barrier function value during each inner iteration. This completes the proof of the theorem. The output gives a ε-approximate solution of the Cartesian P∗(κ)-SCLCP

Iteration bound for a small-update method

Conclusions and remarks