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