Solving A Class of Nonsmooth Resource Allocation Problems with Directed Graphs though Distributed Smooth Multi-Proximal Algorithms

TL;DR

This paper introduces distributed multi-proximal primal-dual algorithms for nonsmooth resource allocation over directed graphs, ensuring convergence to optimal solutions despite communication and nonsmoothness challenges.

Contribution

It proposes novel multi-proximal splitting algorithms that handle unproximable nonsmooth functions and directed graph constraints, with rigorous convergence analysis.

Findings

Algorithms guarantee convergence to optimal resource allocation.

Effective handling of nonsmooth functions in directed graph settings.

Theoretical proof of stability and convergence.

Abstract

In this paper, two distributed multi-proximal primal-dual algorithms are proposed to deal with a class of distributed nonsmooth resource allocation problems. In these problems, the global cost function is the summation of local convex and nonsmooth cost functions, each of which consists of one twice differentiable function and multiple nonsmooth functions. Communication graphs of underling multi-agent systems are directed and strongly connected but not necessarily weighted-balanced. The multi-proximal splitting is designed to deal with the difficulty caused by the unproximable property of the summation of those nonsmooth functions. Moreover, it can also guarantee the smoothness of proposed algorithms. Auxiliary variables in the multi-proximal splitting are introduced to estimate subgradients of nonsmooth functions. Theoretically, the convergence analysis is conducted by employing Lyapunov stability theory and integral input-to-state stability (iISS) theory with respect to set. It shows that proposed algorithms can make states converge to the optimal point that satisfies resource allocation conditions.