Composite Optimization with Coupling Constraints via Penalized Proximal Gradient Method in Partially Asynchronous Networks

TL;DR

This paper introduces an asynchronous penalized proximal gradient algorithm for solving composite optimization problems with linear coupling constraints in multi-agent networks, ensuring convergence despite communication delays.

Contribution

It proposes a novel asynchronous algorithm that handles delays and coupling constraints, with proven convergence rate and practical applications in distributed optimization.

Findings

Convergence rate of O(1/K) under proper conditions

Effective handling of communication delays in multi-agent settings

Successful application to distributed LASSO and electricity market problems

Abstract

In this paper, we consider a composite optimization problem with linear coupling constraints in a multi-agent network. In this problem, all the agents jointly optimize a global composite cost function which is the linear sum of individual cost functions composed of both smooth and non-smooth components. To solve this problem, we propose an asynchronous penalized proximal gradient (Asyn-PPG) algorithm, a variant of classical proximal gradient method, by considering the asynchronous update instants of the agents and communication delays in the network. Specifically, we consider a slot-based asynchronous network (SAN), where the whole time domain is split into sequential time slots and each agent is permitted to make multiple updates during a slot by accessing the historical state information of others. Moreover, we consider a set of global linear constraints and impose some violation penalties on the updating algorithms. By the Asyn-PPG algorithm, we will show that a periodically convergence with rate O(1/K) (K is the index of time slots) can be guaranteed if the coefficient of the penalties for all agents is synchronized at the end of the time slots and the step-size of the Asyn-PPG algorithm is properly determined. The feasibility of the proposed algorithm is verified by solving a consensus based distributed LASSO problem and a social welfare optimization problem in the electricity market respectively.