  We propose a unifying framework for the automated computer-assisted
worst-case analysis of cyclic block coordinate algorithms in the unconstrained
smooth convex optimization setup. We compute exact worst-case bounds for the
cyclic coordinate descent and the alternating minimization algorithms over the
class of smooth convex functions, and provide sublinear upper and lower bounds
on the worst-case rate for the standard class of functions with coordinate-wise
Lipschitz gradients. We obtain in particular a new upper bound for cyclic
coordinate descent that outperforms the best available ones by an order of
magnitude. We also demonstrate the flexibility of our approach by providing new
numerical bounds using simpler and more natural assumptions than those normally
made for the analysis of block coordinate algorithms. Finally, we provide
numerical evidence for the fact that a standard scheme that provably
accelerates random coordinate descent to a $O(1/k^2)$ complexity is actually
inefficient when used in a (deterministic) cyclic algorithm.
