An Inexact Augmented Lagrangian Method for (L0, L1)-Smooth Convex Optimization
Abstract
Augmented Lagrangian methods are among the most effective approaches for solving constrained convex optimization problems. However, classical complexity analyses of first-order methods applied within the augmented Lagrangian framework usually rely on the assumption that the objective function has a Lipschitz continuous gradient. This assumption excludes an important class of generalized smooth functions whose gradients may grow unboundedly. In this paper, we study an inexact augmented Lagrangian method for solving linearly constrained convex optimization problems with (L0,L1)-smooth objective functions. We show that the augmented Lagrangian subproblems preserve the (L0,L1)-smooth structure, with parameters depending on the penalty coefficient. This property allows us to employ recent accelerated first-order schemes designed for generalized smooth optimization instead of classical smooth optimization methods. In particular, we combine the inexact augmented Lagrangian framework with a two-stage acceleration procedure based on clipped gradient descent and accelerated optimization.
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BibTeX
@misc{vyguzov2026inexact,
title = {{An Inexact Augmented Lagrangian Method for $(L_0, L_1)$-Smooth Convex Optimization}},
author = {Aleksandr Vyguzov and Fedor Stonyakin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00261},
eprint = {2609.00261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00261},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Inexact Augmented Lagrangian Method for (L_0, L_1)-Smooth Convex Optimization
AU - Aleksandr Vyguzov
AU - Fedor Stonyakin
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00261
UR - https://arxiv.org/abs/2609.00261
AB - Augmented Lagrangian methods are among the most effective approaches for solving constrained convex optimization problems. However, classical complexity analyses of first-order methods applied within the augmented Lagrangian framework usually rely on the assumption that the objective function has a Lipschitz continuous gradient. This assumption excludes an important class of generalized smooth functions whose gradients may grow unboundedly. In this paper, we study an inexact augmented Lagrangian method for solving linearly constrained convex optimization problems with (L_0,L_1)-smooth objective functions. We show that the augmented Lagrangian subproblems preserve the (L_0,L_1)-smooth structure, with parameters depending on the penalty coefficient. This property allows us to employ recent accelerated first-order schemes designed for generalized smooth optimization instead of classical smooth optimization methods. In particular, we combine the inexact augmented Lagrangian framework with a two-stage acceleration procedure based on clipped gradient descent and accelerated optimization.
ER -