# Primal-Dual Interior-Point Algorithms for Semidefinite Optimization Based on a Simple Kernel Function

## Abstract

Interior-point methods (IPMs) for semidefinite optimization (SDO) have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, J. Peng et al. introduced so-called self-regular kernel (and barrier) functions and designed primal-dual interior-point algorithms based on self-regular proximities for linear optimization (LO) problems. They also extended the approach for LO to SDO. In this paper we present a primal-dual interior-point algorithm for SDO problems based on a simple kernel function which was first presented at the *Proceedings of Industrial Symposium and Optimization Day*, Australia, November 2002; the function is not self-regular. We derive the complexity analysis for algorithms based on this kernel function, both with large- and small-updates. The complexity bounds are
\(\mathrm{O}(qn)\log\frac{n}{\epsilon}\)
and
\(\mathrm{O}(q^{2}\sqrt{n})\log\frac{n}{\epsilon}\)
, respectively, which are as good as those in the linear case.

## Keywords

semidefinite optimization interior-point methods primal-dual methods large- and small-update methods polynomial complexity## Preview

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