  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.
