M511

We develop a novel trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. This class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations inherent in large-scale system solves and compression techniques, e.g. randomized sketching. When using a quadratic Taylor model for the trust-region subproblem, our algorithm is an inexact, matrix-free proximal Newton-type method that permits indefinite Hessians. We prove global convergence of our method in Hilbert space and elaborate on potential nonsmooth subproblem solvers based on ideas taken from their smooth counter-parts. Under additional assumptions, we can also prove superlinear, or even quadratic convergence to local minima. We demonstrate its efficacy on examples from data science and PDE-constrained optimization.

Robert J Baraldi

Sandia National Laboratories