Block Kaczmarz Method with Inequalities

TL;DR

This paper extends the block Kaczmarz method to systems with both equalities and inequalities, showing improved convergence rates through matrix paving and geometric properties, supported by theoretical and experimental results.

Contribution

It introduces a block Kaczmarz method for mixed systems, demonstrating improved convergence via matrix paving and geometric conditions, bridging previous work on equalities and inequalities.

Findings

Matrix paving improves convergence for equalities.

Geometric properties are key for inequalities.

Experimental results support theoretical analysis.

Abstract

The randomized Kaczmarz method is an iterative algorithm that solves overdetermined systems of linear equations. Recently, the method was extended to systems of equalities and inequalities by Leventhal and Lewis. Even more recently, Needell and Tropp provided an analysis of a block version of the method for systems of linear equations. This paper considers the use of a block type method for systems of mixed equalities and inequalities, bridging these two bodies of work. We show that utilizing a matrix paving over the equalities of the system can lead to significantly improved convergence, and prove a linear convergence rate as in the standard block method. We also demonstrate that using blocks of inequalities offers similar improvement only when the system satisfies a certain geometric property. We support the theoretical analysis with several experimental results.