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

Abstract

The -algorithm is a superlinearly convergent method for minimizing nonsmooth, convex functions. At each iteration, the algorithm works with a certain -space and its orthogonal -space, such that the nonsmoothness of the objective function is concentrated on its projection onto the -space, and on the -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 -decompositions. We establish a derivative-free variant of the -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