# Block coordinate descent convergence rate

I have read some literature about the convergence rate of block coordinate descent. They all assume that the object function $f$ is Lipschitz continuous, is there any results for the convergence rate if $f$ is continuous and convex but not have Lipschitz gradient?

Update: Assume the function $f$ is smooth and block coordinate descent can find the minimum of $f$ successfully without getting stuck at some points.

The Wikipedia page has a counterexample: A continuous convex function for which coordinate descent fails to converge but getting stuck in a non-optimal point.

Here are the level lines of this function:

By Nicoguaro - Own work, CC BY 4.0, Link

• I have updated the question. Assume that BCD can find the minimum of $f$, is there any known convergence rate for the problem? – Yi Yang Aug 18 '17 at 10:46
• For some reason I have the feeling that this question is going to change as soon as there appears a new answer... – Dirk Aug 18 '17 at 15:35