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In this paper, the convergence of alternating minimization is established fornon-smooth convex optimization in Banach spaces, and novel rates of convergenceare provided. As objective function a composition of a smooth and a non-smoothpart is considered with the latter being block-separable, e.g., correspondingto convex constraints or regularization. For the smooth part, three differentrelaxations of strong convexity are considered: (i) quasi-strong convexity;(ii) quadratic functional growth; and (iii) plain convexity. Linear convergenceis established for the first two cases, generalizing and improving previousresults for strongly convex problems; sublinear convergence is established forthe third case, also improving previous results from the literature. All theconvergence results have in common, that opposing to previous correspondingresults for the general block coordinate descent, the performance of thealternating minimization is beneficially governed by properties of the singleblocks, instead of global properties. Ultimately, not only the betterconditioned block determines the performance, as has been similarly observed inthe literature. But also the worse conditioned problem enhances the performanceadditionally, resulting in potentially significantly improved convergencerates. Furthermore, by solely using the convexity and smoothness properties ofthe problem, the results immediately apply in general Banach spaces.

On the rate of convergence of alternating minimization for non-smooth non-strongly convex optimization in Banach spaces