Novel Distributed Algorithms Design for Nonsmooth Resource Allocation on Weight-Balanced Digraphs

TL;DR

This paper introduces a novel continuous-time distributed algorithm for nonsmooth resource allocation on weight-balanced digraphs, ensuring convergence to the global optimum even with non-differentiable cost functions.

Contribution

The paper proposes a new distributed algorithm based on gradient descent and projection operators, with proven convergence for nonsmooth convex optimization in networked systems.

Findings

Algorithm converges asymptotically to the global optimum.

Exponential convergence when cost functions are smooth with Lipschitz gradients.

Simulation results validate the effectiveness of the proposed method.

Abstract

In this paper, the distributed resource allocation problem on strongly connected and weight-balanced digraphs is investigated, where the decisions of each agent are restricted to satisfy the coupled network resource constraints and heterogeneous general convex sets. Moreover, the local cost function can be non-smooth. In order to achieve the exact optimum of the nonsmooth resource allocation problem, a novel continuous-time distributed algorithm based on the gradient descent scheme and differentiated projection operators is proposed. With the help of the set-valued LaSalle invariance principle and nonsmooth analysis, it is demonstrated that the algorithm converges asymptotically to the global optimal allocation. Moreover, for the situation where local constraints are not involved and the cost functions are differentiable with Lipschitz gradients, the convergence of the algorithm to the exact optimal solution is exponentially fast. Finally, the effectiveness of the proposed algorithms is illustrated by simulation examples.