# Accelerated Sampling Kaczmarz Motzkin Algorithm for The Linear Feasibility Problem

## TL;DR

This paper introduces an Accelerated Sampling Kaczmarz Motzkin (ASKM) algorithm that improves convergence for large-scale linear feasibility problems, especially in ill-conditioned cases, outperforming existing methods.

## Contribution

The paper proposes a novel accelerated version of the SKM algorithm with proven convergence improvements for solving large-scale linear inequalities.

## Key findings

- ASKM outperforms SKM, IPM, and ASM on various test instances.
- ASKM converges faster on ill-conditioned problems.
- Numerical experiments validate the effectiveness of ASKM.

## Abstract

The Sampling Kaczmarz Motzkin (SKM) algorithm is a generalized method for solving large scale linear systems of inequalities. Having its root in the relaxation method of Agmon, Schoenberg, and Motzkin and the randomized Kaczmarz method, SKM outperforms the state of the art methods in solving large-scale Linear Feasibility (LF) problems. Motivated by SKM's success, in this work, we propose an Accelerated Sampling Kaczmarz Motzkin (ASKM) algorithm which achieves better convergence compared to the standard SKM algorithm on ill conditioned problems. We provide a thorough convergence analysis for the proposed accelerated algorithm and validate the results with various numerical experiments. We compare the performance and effectiveness of ASKM algorithm with SKM, Interior Point Method (IPM) and Active Set Method (ASM) on randomly generated instances as well as Netlib LPs. In most of the test instances, the proposed ASKM algorithm outperforms the other state of the art methods.