A Sampling Kaczmarz-Motzkin Algorithm for Linear Feasibility

TL;DR

This paper introduces a new family of algorithms that blend the relaxation method and randomized Kaczmarz method to efficiently solve large-scale linear inequalities, with proven convergence and improved performance.

Contribution

It presents a novel combined algorithm that generalizes existing methods for linear feasibility, with theoretical convergence guarantees and superior empirical results.

Findings

Algorithms outperform original methods in experiments

Proven convergence of the new algorithms

Effective for large-scale linear inequalities

Abstract

We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. In doing so, we obtain a family of algorithms that generalize and extend both techniques. We prove several convergence results, and our computational experiments show our algorithms often outperform the original methods.