Randomized Methods for Linear Constraints: Convergence Rates and Conditioning

We study randomized variants of two classical algorithms: coordinate descent for systems of linear equations and iterated projections for systems of linear inequalities. Expanding on a recent randomized iterated projection algorithm of Strohmer and Vershynin (Strohmer, T., R. Vershynin. 2009. A randomized Kaczmarz algorithm with exponential convergence. J. Fourier Anal. Appl. 15 262–278) for systems of linear equations, we show that, under appropriate probability distributions, the linear rates of convergence (in expectation) can be bounded in terms of natural linear-algebraic condition numbers for the problems. We relate these condition measures to distances to ill-posedness and discuss generalizations to convex systems under metric regularity assumptions.