Accelerated Golden Ratio Primal--Dual Algorithm for Structured Convex Optimisation without Linesearch

Abstract

This paper revisits the adaptive extended golden-ratio primal--dual algorithm (aEGRPDA) proposed by Soe et al. (2026) for structured convex optimisation problems involving a differentiable term that is only locally smooth. We prove that the artificial upper bound imposed on the primal step-size in aEGRPDA is redundant, since the adaptive rule itself keeps the step-sizes bounded above. As a consequence, the ergodic $\mathcal O(1/N)$ estimates for the objective residual and feasibility violation, where $N\ge1$ denotes the number of iterations, are independent of this hyperparameter. Consequently, the resulting adaptive golden-ratio primal--dual method, therefore, requires neither a step-size cap, nor a linesearch procedure, nor a known global Lipschitz constant. We establish linear convergence of the algorithm when both the primal and dual functions are strongly convex. Furthermore, we develop two accelerated variants, in addition to the local smoothness assumption: one for the case where the nonsmooth primal component is strongly convex, and another for the case where the differentiable term is globally strongly convex. For these accelerated methods, we prove an ergodic $\mathcal O(1/N^2)$ convergence rate. Preliminary numerical experiments on a Poisson imaging problem illustrate the efficiency and robustness of the proposed approaches.