A Nesterov-Accelerated Primal-Dual Splitting Algorithm for Convex Nonsmooth Optimization
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 $\min_x f(x) + g(x) + h(K x)$, accelerating their convergence with respect to the smooth term $f$ is notoriously challenging, due to the rotational dynamics in the primal-dual space. Our guiding principle is that Nesterov acceleration exploits the strong convexity of the dual problem. We first revisit accelerated proximal gradient descent and give a unified analysis of its convex and strongly convex regimes, including the weak convergence of its iterates. We then propose the Accelerated Proximal Alternating Predictor-Corrector algorithm (APAPC) for the setting where $g(x) = (μ_g/2)\|x\|^2$. APAPC is fully split: it only calls $\nabla f$, $K$, $K^*$ and the proximity operator of $h$, and never solves a linear system. From a single Lyapunov inequality, we derive optimal $O(1/t^2)$ convergence rates, as well as accelerated linear rates when $μ_g > 0$, in three regimes where the dual problem is strongly convex: when $h$ is smooth, when $K^*$ is bounded below, and for linearly constrained problems. In the last two regimes, the $O(1/t^2)$ rates without strong convexity of the primal problem are, to our knowledge, the first for fully split algorithms. Finally, leveraging recent results on accelerated gradient descent, we establish the weak convergence of the primal iterates and the strong convergence of the dual iterates to a saddle point.