Trajectory convergence and $o(t^{-2})$ rates for Nesterov accelerated primal-dual dynamics without Lipschitz gradient assumption

TL;DR

This paper proves convergence and improved rates for a Nesterov accelerated primal-dual system solving linearly constrained problems without requiring Lipschitz continuity of the gradient, extending previous results.

Contribution

It establishes convergence and $o(t^{-2})$ rates for the primal-dual dynamics in finite dimensions without Lipschitz gradient assumptions, including the critical case $oldsymbol{ ext{α=3}}$.

Findings

Trajectory converges to a primal-dual solution for $oldsymbol{ ext{α≥3}}$ without Lipschitz gradient.

Improved $o(t^{-2})$ convergence rates for objective residual and feasibility when $oldsymbol{ ext{α>3}}$.

First results on convergence of accelerated primal-dual dynamics for the critical case $oldsymbol{ ext{α=3}}$.

Abstract

We consider the Nesterov accelerated primal-dual dynamical system \[ \begin{cases} \ddot{x}(t)+\dfrac{\alpha}{t}\dot{x}(t) +\nabla f(x(t)) +A^\top\bigl(\lambda(t)+\theta t\dot{\lambda}(t)\bigr)+\beta A^\top(Ax(t)-b)=0,\\[0.6em] \ddot{\lambda}(t)+\dfrac{\alpha}{t}\dot{\lambda}(t) -\bigl(A(x(t)+\theta t\dot{x}(t))-b\bigr)=0, \end{cases} \] which is linked to the linearly constrained optimization problem $ \min_{x\in\mathbb{R}^n} f(x),\ s.t.\ Ax=b, $ where $\alpha\ge 3$ and $f$ is convex and continuously differentiable. In a Hilbert framework, the weak convergence of its trajectory was established by Bo\c{t} and Nguyen (J. Differential Equations, 303:369--406, 2021) under $\alpha>3$ and the Lipschitz continuity assumption on $\nabla f$. In this paper, we prove in finite-dimensional spaces that the trajectory converges to a primal-dual solution for $\alpha\ge3$, without assuming Lipschitz continuity of $\nabla f$. Moreover, when $\alpha>3$, we establish improved $o(t^{-2})$ convergence rates for both the objective residual and the feasibility violation. Our analysis relies on Bregman-distance arguments, instead of the Lipschitz continuity of $\nabla f$. The same strategy can also be extended to time-scaled primal-dual dynamics to obtain analogous convergence results. To the best of our knowledge, this is the first results in this topic without Lipschitz gradient assumption. Our result also present the first work on the convergence of the trajectory of the accelerated primal-dual dynamical system for the critical case $\alpha=3$.