A conditional-gradient-based single-loop augmented Lagrangian method for inequality constrained optimization
Abstract
We consider the problem of minimizing the sum of a Lipschitz differentiable convex function and a proper closed convex function that admits efficient linear minimization oracles, subject to multiple smooth convex inequality constraints. We adapt the classical augmented Lagrangian (AL) method for these problems: in each iteration, our algorithm consists of one step of the conditional gradient (CG) method applied to the AL function, followed by an update of the dual variable as in classical AL methods with a diminishing dual stepsize. We study the convergence rate of our algorithm under two standard stepsize rules for the CG method, namely, an open-loop stepsize and the short stepsize, and obtain a convergence rate that matches the best-known complexity for this class of problems. We also establish accelerated rates when is the indicator function of a uniformly convex set.
Cite
@article{arxiv.2605.22539,
title = {A conditional-gradient-based single-loop augmented Lagrangian method for inequality constrained optimization},
author = {Xiaozhou Wang and Ting Kei Pong and Zev Woodstock},
journal= {arXiv preprint arXiv:2605.22539},
year = {2026}
}