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.
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BibTeX
@misc{baraldi2026convergence,
title = {{Convergence Analysis of an Inexact Proximal Trust-Region Method}},
author = {Robert J. Baraldi and Drew P. Kouri and Leandro Farias Maia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26747},
eprint = {2609.26747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26747},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Convergence Analysis of an Inexact Proximal Trust-Region Method
AU - Robert J. Baraldi
AU - Drew P. Kouri
AU - Leandro Farias Maia
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26747
UR - https://arxiv.org/abs/2609.26747
AB - 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.
ER -