I am using level method to solve non-smooth convex programming problem (where the objective function is given by an oracle from another program ): http://www2.isye.gatech.edu/~nemirovs/Lect_EMCO.pdf chapter 8.2.1

Basically, every iteration a projection of a point into a polytope is needed to be done, and I am using Gurobi (Quadratic programming) to achieve this. However, as my problem dimension could be rather high, say n=5000, an exact projection is slow (takes roughly 20s where as the oracle can be fast taking only 0.5s to compute) and 5000 iterations is needed. I am wondering whether there is way walking around this difficulty, since only a good projection is needed and exact projection is not needed. Thank you.