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

arXiv: https://arxiv.org/abs/2604.09245 · HTML: https://mlsanity.com/paper/2604.09245/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Apr 2026.

## Built on by (most cited first, 1 of 1)

- [Decentralized Optimization with Cross-Coupled Mixed Affine Constraints](https://mlsanity.com/paper/2609.33021/) (2026)

## Abstract (arXiv)

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 minx 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)=fracμg2|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/t2) sublinear convergence rates, as well as accelerated linear convergence when μg > 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.