A unified kernel function approach to primal-dual interior-point algorithms for convex quadratic SDO

Abstract

Kernel functions play an important role in the design and analysis of primal-dual interior-point algorithms. They are not only used for determining the search directions but also for measuring the distance between the given iterate and the μ-center for the algorithms. In this paper we present a unified kernel function approach to primal-dual interior-point algorithms for convex quadratic semidefinite optimization based on the Nesterov and Todd symmetrization scheme. The iteration bounds for large-and small-update methods obtained are analogous to the linear optimization case. Moreover, this unifies the analysis for linear, convex quadratic and semidefinite optimizations.

A Parametric Kernel Function Generating the best Known Iteration Bound for Large-Update Methods for CQSDO

“…Among them, the interior-point methods (IPMs) gained much more attention than others [2,5]. Several, IPMs designed for linear 877 optimization (LO) have been successfully extended to CQSDO, (e.g., [4,14,17,19,22,23,24]), due to their polynomial complexity and practical efficiency.…”

“…We note that similar algorithms are successfully prolonged to convex quadratic optimization over symmetric cone (CQSCO) (see [10,21]). For some other related interior-point algorithms based on the kernel functions we refer to [1,3,6,11,18,22,23,24,25]. A kernel function is a univariate strictly convex function which is defined for all positive real t and is minimal at t = 1 whereas the minimal value equals 0.…”

A Parametric Kernel Function Yielding the Best Known Iteration Bound of Interior-Point Methods for Semidefinite Optimization

“…Many interior-point methods (IPMs) for linear optimization (LO) are successfully extended to (SDO) due to their polynomial complexity and practical efficiency. For an overview of these results, we refer to [1,2] and the references [3,4,5,6,7,8,9,10,11,12,13].…”

A Large-Update Feasible Interior-Point Algorithm for Convex Quadratic Semi-definite Optimization Based on a New Kernel Function

“…He showed that the primal-dual IPMs for solving SDO enjoys O(n Haseli [1] presented an interior-point algorithm for LO based on a new kernel function and obtained the best known iteration bound for large-update methods. Recently, Wang and Zhu [21] proposed a unified kernel function approach to primal-dual IPMs for CQSDO based on kernel functions which were previously introduced for LO in [2]. Later on, Zhang [23] presented a large-update interior-point algorithm for CQSDO based on a new kernel function and obtained its iteration complexity as O( √ n(log n) 2 log n ) for the CQSDO problems.…”