Convergence Analysis of an Inexact Proximal Trust-Region Method

Abstract

We analyze an inexact, proximal trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function. Our algorithm leverages the proximal gradient as a benchmark to guarantee global convergence. However, evaluating the proximal gradient involves solving a nonsmooth convex optimization problem that in many applications cannot be performed exactly. As such, we develop a framework that permits inexact evaluations of the proximity operator and we show the proposed inexactness conditions yield descent properties analogous to the exact proximal gradient. We further discuss practical procedures for computing approximate proximity operators for two common applications: (i) the nonsmooth term is the sum of nonsmooth convex functions where one is composed with an affine map and (ii) using alternative inner products to facilitate the evaluation of the proximity operator. We demonstrate our algorithm and confirm our analysis with two numerical experiments arising in PDE-constrained optimization.