A distributed semismooth Newton based augmented Lagrangian method for distributed optimization

TL;DR

This paper introduces a novel distributed optimization algorithm combining semismooth Newton and augmented Lagrangian methods, achieving efficient convergence and communication efficiency over networks.

Contribution

It develops a distributed semismooth Newton method integrated with an augmented Lagrangian approach, with a new accelerated proximal gradient technique for computing Newton directions.

Findings

Demonstrates faster convergence than existing methods

Reduces communication costs in distributed settings

Proves theoretical convergence guarantees

Abstract

This paper proposes a novel distributed semismooth Newton based augmented Lagrangian method for solving a class of optimization problems over networks, where the global objective is defined as the sum of locally held cost functions, and communication is restricted to neighboring agents. Specifically, we employ the augmented Lagrangian method to solve an equivalently reformulated constrained version of the original problem. Each resulting subproblem is solved inexactly via a distributed semismooth Newton method. By fully leveraging the structure of the generalized Hessian, a distributed accelerated proximal gradient method is proposed to compute the Newton direction efficiently, eliminating the need to communicate with full Hessian matrices. Theoretical results are also obtained to guarantee the convergence of the proposed algorithm. Numerical experiments demonstrate the efficiency and superiority of our algorithm compared to state-of-the-art distributed algorithms.