SFU MOCAD Seminar: Robert John Baraldi

We introduce an inexact trust-region method for efficiently solving a class of problems in which the objective is the sum of a smooth, nonconvex function and nonsmooth, convex function. Such objectives are pervasive in the literature, with examples being machine learning, basis pursuit, inverse problems, and topology optimization. The inclusion of nonsmooth regularizers and constraints is critical, as they often preserve physical properties or promote sparsity in the control. Enforcing these properties in an efficient manner is critical when met with computationally intense nature of solving PDEs or machine learning applications. We develop a novel trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations. 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. Moreover, we provide extensions of this method to adaptive mesh refinement, stochastic optimization as well as multilevel procedures.We prove global convergence of our method in Hilbert space and demonstrate its efficacy on examples from data science and PDE-constrained optimization.