A Mizuno-Todd-Ye predictor-corrector infeasible-interior-point method for symmetric optimization with the arc-search strategy

Ximei Yang,Yinkui Zhang

doi:10.1186/s13660-017-1565-y

Ximei Yang, Yinkui Zhang

Open Access

https://doi.org/10.1186/s13660-017-1565-y

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Abstract

In this paper, we propose a Mizuno-Todd-Ye predictor-corrector infeasible-interior-point method for symmetric optimization using the arc-search strategy. The proposed algorithm searches for optimizers along the ellipses that approximate the central path and ensures that the duality gap and the infeasibility have the same rate of decline. By analyzing, we obtain the iteration complexity mathcal{O}(rlog varepsilon^{-1}) for the Nesterov-Todd direction, where r is the rank of the associated Euclidean Jordan algebra and ε is the required precision. To our knowledge, the obtained complexity bounds coincide with the currently best known theoretical complexity bounds for infeasible symmetric optimization.

Highlights

1 Introduction The purpose of this paper is to propose a Mizuno-Todd-Ye predictor-corrector (MTY-PC) infeasible-interior-point method for symmetric optimization (SO) by using Euclidean Jordan algebra (EJA)
In the nineties of the last century, researchers began to focus on the MTY-PC algorithm [ – ], because it had the property of the best iteration complexity obtained so far for all the interior-point method (IPM)
In order to further study the advantages of the arc-search algorithm, Yang [, ] proposed two infeasible-IPMs for linear optimization (LO) and SO, and respectively obtained the O(n / log ε– )-iteration complexity for LO and the O(r / log ε– ) and O(r / log ε– )-iteration complexity, where n is the larger dimension of a standard LO, r is the rank of the associated EJA and ε is the required precision

Summary

Introduction

The purpose of this paper is to propose a Mizuno-Todd-Ye predictor-corrector (MTY-PC) infeasible-interior-point method (infeasible-IPM) for symmetric optimization (SO) by using Euclidean Jordan algebra (EJA). In order to further study the advantages of the arc-search algorithm, Yang [ , ] proposed two infeasible-IPMs for LO and SO, and respectively obtained the O(n / log ε– )-iteration complexity for LO and the O(r / log ε– ) and O(r / log ε– )-iteration complexity, where n is the larger dimension of a standard LO, r is the rank of the associated EJA and ε is the required precision. A complete system of orthogonal primitive idempotents is called a Jordan frame. Systems ( ) and ( ) do not always have a unique solution due to the fact that x and s do not operator commute in general To overcome this difficulty, we apply a scaling scheme that follows from [ , Lemma ].

The lower bounds on θ

Conclusion