A universal accelerated primal-dual method for convex optimization problems

TL;DR

This paper introduces a universal accelerated primal-dual algorithm for convex optimization that adapts to different gradient smoothness levels without prior knowledge, achieving optimal convergence rates.

Contribution

It develops a new universal primal-dual method that handles both Lipschitz and Hölder gradients with adaptive line search, providing optimal convergence without smoothness assumptions.

Findings

Achieves universal optimal convergence rates.

Handles both Lipschitz and Hölder gradient cases.

Confirmed efficiency through numerical experiments.

Abstract

This work presents a universal accelerated first-order primal-dual method for affinely constrained convex optimization problems. It can handle both Lipschitz and H\"{o}lder gradients but does not need to know the smoothness level of the objective function. In line search part, it uses dynamically decreasing parameters and produces approximate Lipschitz constant with moderate magnitude. In addition, based on a suitable discrete Lyapunov function and tight decay estimates of some differential/difference inequalities, a universal optimal mixed-type convergence rate is established. Some numerical tests are provided to confirm the efficiency of the proposed method.