Regularized extragradient method for structured bilevel optimization in continuous and discrete time
Abstract
In a real Hilbert space, we study a bilevel optimization problem that consists in minimizing an outer convex function over the zero set of a maximally monotone operator. In the smooth setting, where the outer objective is convex and Fréchet differentiable and the inner operator is single-valued, continuous and monotone, we associate with the problem a first-order dynamical system that can be viewed as a monotone flow applied to a dynamically regularized operator. Under a suitable Attouch-Czarnecki-type condition for the inner level, we establish last-iterate convergence rates for both the outer and inner residuals, together with weak convergence of the trajectories to optimal solutions of the bilevel problem. In the smooth+nonsmooth setting, we enrich the outer objective with a proper, convex, and lower semicontinuous function, while the inner operator is augmented by the subdifferential of a function with the same properties. We propose a regularized proximal-extragradient algorithm in which both the forward and backward steps are performed with respect to dynamically regularized operators and functions, respectively. Under an analogous Attouch-Czarnecki-type condition formulated for the smooth+nonsmooth case, we establish last-iterate convergence rates for both the outer and inner residuals, together with weak convergence of the iterates to optimal solutions of the bilevel problem.
Related
- A Stochastic Proximal-Gradient Method for Regularized Generally Constrained Continuous Optimization
- A Bregman inertial iteratively regularized extragradient method for bilevel variational inequality problems
- Inertial Primal-Dual Dynamics Methods Featuring Implicit Hessian-Driven Damping for Convex Optimization Problems in Continuous and Discrete Time
- Near-Optimal Single-Loop Predictor--Corrector Extragradient Method for Strongly Convex--Strongly Concave Minimax Optimization
- Single-Loop Stochastic Projected Damped Extragradient Methods for Nonconvex--(Strongly) Concave Minimax Optimization
BibTeX
@misc{bot2026regularized,
title = {{Regularized extragradient method for structured bilevel optimization in continuous and discrete time}},
author = {Radu Ioan Bot and Enis Chenchene and David Alexander Hulett},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29181},
eprint = {2608.29181},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29181},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regularized extragradient method for structured bilevel optimization in continuous and discrete time
AU - Radu Ioan Bot
AU - Enis Chenchene
AU - David Alexander Hulett
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2608.29181
UR - https://arxiv.org/abs/2608.29181
AB - In a real Hilbert space, we study a bilevel optimization problem that consists in minimizing an outer convex function over the zero set of a maximally monotone operator. In the smooth setting, where the outer objective is convex and Fréchet differentiable and the inner operator is single-valued, continuous and monotone, we associate with the problem a first-order dynamical system that can be viewed as a monotone flow applied to a dynamically regularized operator. Under a suitable Attouch-Czarnecki-type condition for the inner level, we establish last-iterate convergence rates for both the outer and inner residuals, together with weak convergence of the trajectories to optimal solutions of the bilevel problem. In the smooth+nonsmooth setting, we enrich the outer objective with a proper, convex, and lower semicontinuous function, while the inner operator is augmented by the subdifferential of a function with the same properties. We propose a regularized proximal-extragradient algorithm in which both the forward and backward steps are performed with respect to dynamically regularized operators and functions, respectively. Under an analogous Attouch-Czarnecki-type condition formulated for the smooth+nonsmooth case, we establish last-iterate convergence rates for both the outer and inner residuals, together with weak convergence of the iterates to optimal solutions of the bilevel problem.
ER -