A primal-dual large-update interior-point algorithm for semi-definite optimization based on a new kernel function

Dequan Zhao,Mingwang Zhang

doi:10.19139/soic.v1i1.8

Abstract

Based on a new parametric kernel function, this paper presents a primal-dual large-update interior-point algorithm (IPM) for semi-definite optimization (SDO) problems. The new parametric function is neither self-regular function nor the usual logarithmic barrier function. It is strongly convex and possesses some novel analytic properties. We analyse this new parametric kernel function and show that the proposed algorithm has favorable complexity bound in terms of the analytic properties of the kernel function. Moreover, the complexity bound for our large-update IPM is shown to be O(\sqrt{n}(\log n)^2 \log\frac{n}{\epsilon}). Some numerical results are reported to illustrate the feasibility of the proposed algorithm.

Highlights

Throughout this paper we deal with complexity analysis of large-update primaldual interior-point algorithm (IPM) for solving semi-definite optimization (SDO) problems
Motivated by [16], in this paper we propose a primal-dual large-update IPM for SDO based on the kernel function: ψ(t)
We do some analysis about our new parametric kernel function ψ(t) and design the corresponding primal-dual IPM based on ψ(t)

Summary

Introduction

Throughout this paper we deal with complexity analysis of large-update primaldual IPMs for solving SDO problems. J.Peng et al [9, 10] introduced so-called self-regular kernel functions and proposed primal-dual IPM for LO based on self-regular function, and extended currently the the approach to SDO. Bai et al [1] developed a class of primal-dual IPMs for LO based on elibigle barrier function and obtained the same favorable iteration bounds for the algorithms [9]. [16] developed some new analysis tools for convex quadratic sOe(m√i-nd(elfiogninte)2olpotgimnε i)z.aTtihoen based bound on is a new kernel function better than that by the with iteration bound classical primal-dual. Motivated by [16], in this paper we propose a primal-dual large-update IPM for SDO based on the kernel function: ψ(t) t2. We do some analysis about our new parametric kernel function ψ(t) and design the corresponding primal-dual IPM based on ψ(t).

Preliminaries

A generic primal-dual algorithm for SDO

Analysis of the Algorithm for SDO

Numerical Results

Concluding Remarks