A Nesterov-Accelerated Primal-Dual Splitting Algorithm for Convex Nonsmooth Optimization

TL;DR

This paper introduces an accelerated primal-dual splitting algorithm that incorporates Nesterov momentum, achieving optimal convergence rates for structured convex nonsmooth problems by leveraging strong convexity and a unified Lyapunov analysis.

Contribution

It proposes the APAPC algorithm integrating Nesterov acceleration into primal-dual methods for convex optimization, with proven optimal convergence rates and stability analysis.

Findings

Achieves $O(1/t^2)$ sublinear convergence rate.

Establishes accelerated linear convergence under strong convexity.

Provides weak convergence guarantees for primal-dual iterates.

Abstract

We investigate the integration of Nesterov-type acceleration into primal-dual methods for structured convex optimization. While proximal splitting algorithms efficiently handle composite problems of the form $\min_x f(x)+g(x)+h(Kx)$, accelerating their convergence with respect to the smooth term $f$ is notoriously challenging due to the rotational dynamics in the primal-dual space. In this paper, we overcome this barrier by proposing the Accelerated Proximal Alternating Predictor-Corrector algorithm (APAPC), focusing on the setting where $g(x)=\frac{\mu_g}{2}\|x\|^2$. Our analysis reveals that Nesterov momentum can be seamlessly integrated into a primal-dual forward-backward scheme by exploiting the strong convexity of the dual problem to stabilize the accelerated primal updates. Using a unified Lyapunov framework, we establish optimal $O(1/t^2)$ sublinear convergence rates, as well as accelerated linear convergence when $\mu_g > 0$, across three regimes of dual strong convexity: (i) when $h$ is smooth, (ii) when the linear operator $K^*$ is bounded below, and (iii) for linearly constrained optimization. Furthermore, leveraging recent results on accelerated gradient descent, we characterize the weak convergence of the primal-dual iterates to a saddle-point solution.