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

Abstract

In this paper, we design an inertial accelerated primal–dual algorithm to address the convex–concave saddle point problem, which is formulated as ${min}_x{max}_{y}f\left(x\right)+〈Kx,y〉-g\left(y\right)$. 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 $O(1/k^2)$ convergence rate, where $k$ represents the number of iterations. Several numerical experiments validate the efficiency of our proposed algorithm.