Primal-Dual Entropy Based Interior-Point Algorithms for Linear Optimization

TL;DR

This paper introduces a new family of primal-dual entropy-based search directions for interior-point methods in linear optimization, achieving optimal iteration complexity and improving search direction selection through plane search algorithms.

Contribution

It develops entropy-based search directions within interior-point methods and proposes heuristic and exact plane search algorithms to optimize step size and direction.

Findings

Achieves the best known iteration complexity bound for linear optimization.

Demonstrates the effectiveness of entropy-based search directions through computational experiments.

Shows that plane search algorithms improve the selection of search directions and step sizes.

Abstract

We propose a family of search directions based on primal-dual entropy in the context of interior-point methods for linear optimization. We show that by using entropy based search directions in the predictor step of a predictor-corrector algorithm together with a homogeneous self-dual embedding, we can achieve the current best iteration complexity bound for linear optimization. Then, we focus on some wide neighborhood algorithms and show that in our family of entropy based search directions, we can find the best search direction and step size combination by performing a plane search at each iteration. For this purpose, we propose a heuristic plane search algorithm as well as an exact one. Finally, we perform computational experiments to study the performance of entropy-based search directions in wide neighborhoods of the central path, with and without utilizing the plane search algorithms.