Block Kaczmarz Method with Inequalities

Abstract

The randomized Kaczmarz method is an iterative algorithm that solves systems of linear equations. Recently, the randomized 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 this randomized 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.