TITLE: Convex optimization with full subdifferential information
QUESTION [5 upvotes]: Can anyone direct me to any algorithms or theorems that describe the difficulty of solving a non-smooth convex optimization problem for the special case where the full subdifferential is available?  All of the results that I can find are designed for the case where one has an oracle that can generate a single subgradient.

REPLY [4 votes]: Regarding difficulty results (i.e., lower bounds) for iterative methods, worst-case complexity results in the oracle model hold regardless of whether you provide the full subgradient to the algorithm. What happens is that lower bounds for nonsmooth convex optimization are obtained by piecewise linear functions, and the adversary can always (consistently) adapt the instance so that at all query points the function is locally linear, thus the subdifferential is a singleton. You can see the proof at page 120 in the following notes http://www2.isye.gatech.edu/~nemirovs/Lect_EMCO.pdf
If you care about average-case analysis (where the adversary is not allowed to adapt) then the question is much more delicate, and I think there is no answer in the literature up to date.