Seminar in hybrid format in GERAD room 4488 or Zoom.

Many applications require minimizing the sum of smooth and nonsmooth functions. For example, basis pursuit denoising problems in data science require minimizing a measure of data misfit plus an \(\ell^1\)-regularizer. Similar problems arise in optimal control of partial differential equations (PDEs) when sparsity of the control is desired. 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 systemically 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 permits indefinite Hessians. We prove global convergence of our method in Hilbert space and demonstrate its efficacy on examples from PDE-constrained optimization.