TY - JOUR
T1 - Convergence of a Generalized Primal-Dual Algorithm with an Improved Condition for Saddle Point Problems
AU - Jiang , Fan
AU - Luo , Yueying
AU - Cai , Xingju
AU - Wang , Tanxing
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 463
EP - 486
PY - 2025
DA - 2025/05
SN - 18
DO - http://doi.org/10.4208/nmtma.OA-2024-0105
UR - https://global-sci.org/intro/article_detail/nmtma/24072.html
KW - Generalized primal-dual algorithm, saddle point problem, convex programming, convergence rate.
AB - <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>