Accelerated linearized alternating direction method of multipliers with Nesterov extrapolation

TL;DR

This paper introduces accelerated linearized ADMM algorithms with Nesterov extrapolation for convex optimization problems, achieving faster convergence rates under certain conditions, and extending classical acceleration methods to constrained problems.

Contribution

The paper develops two accelerated linearized ADMM variants with Nesterov extrapolation for linearly constrained convex problems, extending classical methods and achieving optimal convergence rates.

Findings

Achieve non-ergodic convergence rate of O(1/k^2) under strong convexity.

Reduce to accelerated linearized ALMs for one-block problems.

Numerical results confirm improved efficiency and reliability.

Abstract

The alternating direction method of multipliers (ADMM) has found widespread use in solving separable convex optimization problems. In this paper, by employing Nesterov extrapolation technique, we propose two families of accelerated linearized ADMMs for addressing two-block linearly constrained separable convex optimization problems where each block of the objective function exhibits a ``nonsmooth'' + ``smooth'' composite structure. Our proposed accelerated linearized ADMMs extend two classical Nesterov acceleration methods designed for unconstrained composite optimization problems to linearly constrained problems. These methods are capable of achieving non-ergodic convergence rates of $\mathcal{O}(1/k^2)$, provided that one block of the objective function exhibits strong convexity and the gradients of smooth terms are Lipschitz continuous. We show that the proposed methods can reduce to accelerated linearized augmented Lagrangian methods (ALMs) for solving one-block linearly constrained convex optimization problems. By choosing different extrapolation parameters, we explore the relationship between the proposed methods and some existing accelerated methods. Numerical results are presented to validate the efficacy and reliability of the proposed algorithms.