Symmetry is a recurring feature in algorithms for monotone operator theory and convex optimization, particularly in problems involving the sum of two operators, as exemplified by the Peaceman–Rachford splitting scheme. However, in more general settings—such as composite optimization problems with three convex functions or structured convex-concave saddle-point formulations—existing algorithms often exhibit inherent asymmetry. In particular, the Condat–V{\~u} algorithm and the asymmetry forward-backward-adjoint (AFBA) method, while efficient and widely adopted, apply extrapolation only to either the primal or the dual update, resulting in unbalanced iterations. In this work, we introduce a symmetric primal-dual algorithm (SPDA) that applies extrapolation to both primal and dual iterates, thereby preserving symmetry in the iteration scheme. The algorithm encompasses the Condat–V{\~u} and AFBA methods as special cases and permits more flexible step-size choices. We establish global convergence under standard assumptions and derive both ergodic and non-ergodic convergence rates. The results demonstrate that symmetry can be preserved in first-order methods for optimizing the sum of three convex functions without compromising convergence guarantees or practical simplicity
Citation
Lihan Zhou, Feng Ma. A Symmetric Primal-Dual method with two extrapolation steps for Composite Convex Optimization [J]. Avaliable on http://www.optimization-online.org, 2025.