A Class of Large-Update and Small-Update Primal-Dual Interior-Point Algorithms for Linear Optimization

Abstract

In this paper we present a class of polynomial primal-dual interior-point algorithms for linear optimization based on a new class of kernel functions. This class is fairly general and includes the classical logarithmic function, the prototype selfregular function, and non-self-regular kernel functions as special cases. The analysis of the algorithms in the paper follows the same line of arguments as in Bai et al. (SIAM J. Optim. 15:101-128, 2004), where a variety of non-self-regular kernel functions were considered including the ones with linear and quadratic growth terms. However, the important case when the growth term is between linear and quadratic was not considered. The goal of this paper is to introduce such class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. They match the currently best known iteration bounds for the prototype self-regular function with quadratic growth term, the simple nonself-regular function with linear growth term, and the classical logarithmic kernel function. In order to achieve these complexity results, several new arguments had to be used.

Unified Analysis of Kernel-Based Interior-Point Methods for $P_*(\kappa)$-Linear Complementarity Problems

“…After a step size α is determined, the new iterate (x + , s + ) is calculated from (7). Recall that during an inner iteration the parameter μ is fixed.…”

“…Subsequently, Bai, El Ghami, and Roos [5] and Bai et al [7] presented primal-dual IPMs for LO based on so-called eligible kernel functions, which are not necessarily self-regular. For some of them they matched the best iteration bounds for largeupdate IPMs.…”

“…They managed to extend the results in [5] to semidefinite and second order cone optimization problems [27,8]. Furthermore, the results for LO in [7], which were based on a specific eligible kernel function, were extended to P * (κ)-LCPs in Bai, Lesaja, and Roos [6], again matching the best known iteration bounds with the addition of a factor that depends affinely on κ.…”

Kernel-Based Interior-Point Methods for Monotone Linear Complementarity Problems over Symmetric Cones

“…Function ψ 10 includes functions ψ 1 , ψ 4 , ψ 7 , ψ 8 and ψ 9 as special cases. It was first considered in [38]. …”

Iteration Bounds of Interior-Point Methods for Linear Optimization Based on a New Kernel Function

“…Then, Bai et al [9][10][11][12][13][14] proposed new primal-dual IPMs based on various kernel functions to improve the iteration bounds for large-update methods from ( ) …”