A Lyapunov Analysis of Accelerated PDHG Algorithms

The primal-dual hybrid gradient (PDHG) algorithm is a prominent first-order primal-dual method designed to efficiently solve saddle point problems and associated convex optimization problems. Recently, the underlying mechanism of the PDHG algorithm has been elucidated through the high-resolution ordinary differential equation (ODE) and the implicit-Euler scheme as detailed in [20]. This insight has spurred the development of several accelerated variants of the PDHG algorithm, originally proposed by [4]. By employing discrete Lyapunov analysis, we establish that the PDHG algorithm with iteration-varying step sizes, converges at a rate near $$O(1/k^2)$$ O ( 1 / k 2 ) . Furthermore, for the specific setting where $$\tau _{k+1}\sigma _k = s^2$$ τ k + 1 σ k = s 2 and $$\theta _k = \tau _{k+1}/\tau _k \in (0, 1)$$ θ k = τ k + 1 / τ k ∈ ( 0 , 1 ) as proposed in [4], an even faster convergence rate of $$O(1/k^2)$$ O ( 1 / k 2 ) can be achieved. To substantiate these findings, we design a novel discrete Lyapunov function. This function is distinguished by its succinctness and straightforwardness, providing a clear and elegant proof of the enhanced convergence properties of the PDHG algorithm under the specified conditions. Finally, we utilize the discrete Lyapunov function to establish the optimal linear convergence rate when both the objective functions are strongly convex. The theoretical convergence results are validated by numerical experiments.