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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.

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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:

Nonsmooth coordinate descent.svg
By Nicoguaro - Own work, CC BY 4.0, Link

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  • $\begingroup$ I have updated the question. Assume that BCD can find the minimum of $f$, is there any known convergence rate for the problem? $\endgroup$
    – JYY
    Commented Aug 18, 2017 at 10:46
  • $\begingroup$ For some reason I have the feeling that this question is going to change as soon as there appears a new answer... $\endgroup$
    – Dirk
    Commented Aug 18, 2017 at 15:35

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