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Unifying Variational View of Accelerated Primal-Dual Methods
Abstract
We develop a unifying variational framework for accelerated primal-dual flows in affinely constrained convex optimization. In particular, it is shown that applying the Euler-Lagrange-Rayleigh (ELR) principle to coupled augmented Bregman Lagrangians systematically generates accelerated dynamics over general Bregman geometries and recovers a family of existing primal-dual mirror and Alternating Direction Method of Multipliers (ADMM) flows as special cases. We then establish 𝒪(e−bt) convergence guarantees for these flows under mild assumptions. Lastly, we show that the proposed framework for algorithmic development extends from finite-dimensional optimization to constrained optimization over probability distributions.
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BibTeX
@misc{chang2026unifying,
title = {{Unifying Variational View of Accelerated Primal-Dual Methods}},
author = {Cheng Chang and Mehran Mesbahi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09454},
eprint = {2609.09454},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09454},
}
RIS
TY - UNPB
TI - Unifying Variational View of Accelerated Primal-Dual Methods
AU - Cheng Chang
AU - Mehran Mesbahi
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.09454
UR - https://arxiv.org/abs/2609.09454
AB - We develop a unifying variational framework for accelerated primal-dual flows in affinely constrained convex optimization. In particular, it is shown that applying the Euler-Lagrange-Rayleigh (ELR) principle to coupled augmented Bregman Lagrangians systematically generates accelerated dynamics over general Bregman geometries and recovers a family of existing primal-dual mirror and Alternating Direction Method of Multipliers (ADMM) flows as special cases. We then establish 𝒪(e^−b_t) convergence guarantees for these flows under mild assumptions. Lastly, we show that the proposed framework for algorithmic development extends from finite-dimensional optimization to constrained optimization over probability distributions.
ER -