A New Efficient Large-Update Primal-Dual Interior-Point Method Based on a Finite Barrier

Abstract

We introduce a new barrier function which is not a barrier function in the usual sense: it has finite value at the boundary of the feasible region. Despite this, its iteration bound, O √ n log n log n ε , is as good as it can be: it is the best known bound for large-update methods. The recently introduced notions of superconvexity and exponential convexity are crucial in the analysis.

NEW COMPLEXITY ANALYSIS OF PRIMAL-DUAL IMPS FOR P*LAPS BASED ON LARGE UPDATES

“…√ n log n log n ε ) complexity which is similar to the one in [1] which is so far the best known complexity for large-update IPM. In [1], they analyzed the complexity bound of the algorithm for LO based on all eligible conditions.…”

“…We denote the solution of (2) as (x(µ), s(µ)) for given µ > 0. We also call it µ-center for given µ and the solution set {(x(µ), s(µ)) | µ > 0} the central path for system (1). Note that the sequence (x(µ), s(µ)) approaches to the solution (x, s) of the system (1) as µ → 0 ( [7]).…”

“…In this paper we generalized the large-update primal-dual interior point algorithm ( [1]) for LO to P * LCP. The analysis in this paper uses the analysis scheme presented in [2].…”

“…In [1], they analyzed the complexity bound of the algorithm for LO based on all eligible conditions. But we used three eligible conditions.…”

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

“…Following the terminology introduced in [5][6][7], we call the univariate function ψ(t) the kernel function of the barrier function (v). Obviously, in the case (6) to find x and s. They can also be obtained directly by solving the following system:…”