@Article{NMTMA-18-463,
author = {Jiang , FanLuo , YueyingCai , Xingju and Wang , Tanxing},
title = {Convergence of a Generalized Primal-Dual Algorithm with an Improved Condition for Saddle Point Problems},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2025},
volume = {18},
number = {2},
pages = {463--486},
abstract = {<p style="text-align: justify;">We consider a general convex-concave saddle point problem that frequently arises in large-scale image processing. First-order primal-dual algorithms
have garnered significant attention due to their promising results in solving saddle point problems. Notably, these algorithms exhibit improved performance with
larger step sizes. In a recent series of articles, the upper bound on step sizes has
been increased, thereby relaxing the convergence-guaranteeing condition. This paper analyzes the generalized primal-dual method proposed in [B. He, F. Ma, S. Xu,
X. Yuan, SIAM J. Imaging Sci. 15 (2022)] and introduces a better condition to ensure its convergence. This enhanced condition also encompasses the optimal upper
bound of step sizes in the primal-dual hybrid gradient method. We establish both
the global convergence of the iterates and the ergodic $\mathcal{O}(1/N)$ convergence rate
for the objective function value in the generalized primal-dual algorithm under the
enhanced condition.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2024-0105},
url = {http://global-sci.org/intro/article_detail/nmtma/24072.html}
}