A derivative-free 𝒱𝒰-algorithm for convex finite-max problems. (3rd May 2020) - Hare, Warren; Planiden, Chayne; Sagastizábal, Claudia
ABSTRACT: The V U -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 V U -decompositions. We establish a derivative-free variant of the V U -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.