A Proximal Gradient Method for Nonconvex Nonsmooth Optimization with Applications
Abstract
We study a proximal-gradient method for the sum of a convex smooth function g and a possibly nonconvex, nonsmooth, strongly quasar-convex regularizer. The gradient step is weighted by the quasar-convexity parameter, so, for step sizes in the descent range, the iteration decreases a weighted objective, while the unweighted sum serves for comparison. We establish a geometric envelope around every global minimizer of the unweighted objective, with an explicit radius independent of the step size; a common minimizer yields global linear convergence. We also prove that every critical point u of the weighted objective lies within an explicit multiple of |∇ g(u)| from every global minimizer of the unweighted objective. Together with an optimized composite comparison, this yields asymptotic distance bounds. An example shows that the two objectives can have different critical points. For power-regularized least squares with powers between zero and one, descent provides a confinement ball for the sharp local modulus, and support identification combined with a KL argument yields whole-sequence convergence.
Related
- Asymptotic Analysis of Gradient Mapping-type Stationarity Measure for the Sum of Nonconvex Nonsmooth Functions and Applications to Proximal Gradient-type Algorithms
- Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization
- Controlled Block Proximal Linearized Method for Nonconvex and Nonsmooth Optimization
- Generalized Geometry Block Proximal Linearized Method for Multiblock Nonconvex and Nonsmooth Optimization
- The Complexity of Finding Stationary Points in Nonsmooth Nonconvex Optimization
BibTeX
@misc{brito2026proximal,
title = {{A Proximal Gradient Method for Nonconvex Nonsmooth Optimization with Applications}},
author = {Jose M. M. Brito and Felipe Lara and Yinjun Yuan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32750},
eprint = {2610.32750},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32750},
note = {CC0},
}
RIS
TY - UNPB
TI - A Proximal Gradient Method for Nonconvex Nonsmooth Optimization with Applications
AU - Jose M. M. Brito
AU - Felipe Lara
AU - Yinjun Yuan
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32750
UR - https://arxiv.org/abs/2610.32750
AB - We study a proximal-gradient method for the sum of a convex smooth function g and a possibly nonconvex, nonsmooth, strongly quasar-convex regularizer. The gradient step is weighted by the quasar-convexity parameter, so, for step sizes in the descent range, the iteration decreases a weighted objective, while the unweighted sum serves for comparison. We establish a geometric envelope around every global minimizer of the unweighted objective, with an explicit radius independent of the step size; a common minimizer yields global linear convergence. We also prove that every critical point u of the weighted objective lies within an explicit multiple of |∇ g(u)| from every global minimizer of the unweighted objective. Together with an optimized composite comparison, this yields asymptotic distance bounds. An example shows that the two objectives can have different critical points. For power-regularized least squares with powers between zero and one, descent provides a confinement ball for the sharp local modulus, and support identification combined with a KL argument yields whole-sequence convergence.
ER -