Distributed Continuous-Time Algorithm for Constrained Convex Optimizations via Nonsmooth Analysis Approach

TL;DR

This paper introduces a novel distributed continuous-time algorithm for solving constrained nonsmooth convex optimization problems, ensuring agents reach a consensus on the optimal solution while maintaining bounded states, using nonsmooth Lyapunov stability analysis.

Contribution

It proposes a new distributed continuous-time projected algorithm for constrained nonsmooth convex optimization with convergence guarantees and boundedness of states.

Findings

All agents reach the same optimal solution.

States remain bounded during optimization.

Convergence proven via nonsmooth Lyapunov functions.

Abstract

This technical note studies the distributed optimization problem of a sum of nonsmooth convex cost functions with local constraints. At first, we propose a novel distributed continuous-time projected algorithm, in which each agent knows its local cost function and local constraint set, for the constrained optimization problem. Then we prove that all the agents of the algorithm can find the same optimal solution, and meanwhile, keep the states bounded while seeking the optimal solutions. We conduct a complete convergence analysis by employing nonsmooth Lyapunov functions for the stability analysis of differential inclusions. Finally, we provide a numerical example for illustration.