Non-ergodic convergence rate of an inertial accelerated primal-dual algorithm for saddle point problems

In this paper, we design an inertial accelerated primal-dual algorithm to address the convex-concave saddle point problem, which is formulated as $\min_{x}\max_{y} f(x) + \langle Kx, y \rangle - g(y)$. Remarkably, both functions $f$ and $g$ exhibit a composite structure, combining ``nonsmooth'' + ``smooth'' components. Under the assumption of partially strong convexity in the sense that $f$ is convex and $g$ is strongly convex, we introduce a novel inertial accelerated primal-dual algorithm based on Nesterov's extrapolation. This algorithm can be reduced to two classical accelerated forward-backward methods for unconstrained optimization problem. We show that the proposed algorithm achieves a non-ergodic $\mathcal{O}(1/k^2)$ convergence rate, where $k$ represents the number of iterations. Several numerical experiments validate the efficiency of our proposed algorithm.