A derivative-free VU-algorithm for convex finite-max problems

The VU-algorithm is a superlinearly convergent method for minimizing nonsmooth, convex functions. At each iteration, the algorithm works with a certain V-space and its orthogonal u-space, such that the nonsmoothness of the objective function is concentrated on its projection onto the V-space, and on the U-space the projection is smooth. This structure allows for an alternation between a Newton-like step where the function is smooth, and a proximal-point step that is used to find iterates with promising VU-decompositions. We establish a derivative-free variant of the VU-algorithm for convex finite-max objective functions. We show global convergence and provide numerical results from a proof-of-concept implementation, which demonstrates the feasibility and practical value of the approach. We also carry out some tests using nonconvex functions and discuss the results.