Distributed Nonsmooth Optimization with Coupled Inequality Constraints via Modified Lagrangian Function

TL;DR

This paper introduces a distributed continuous-time algorithm for convex optimization problems with nonsmooth costs and coupled inequality constraints, ensuring convergence through nonsmooth analysis and Lyapunov methods.

Contribution

It proposes a novel modified Lagrangian function with local multipliers and a nonsmooth penalty, along with a distributed primal-dual algorithm for nonsmooth convex problems.

Findings

Algorithm converges under nonsmooth analysis.

Existence of solutions established via Lyapunov functions.

Applicable to distributed convex optimization with coupled constraints.

Abstract

This technical note considers a distributed convex optimization problem with nonsmooth cost functions and coupled nonlinear inequality constraints. To solve the problem, we first propose a modified Lagrangian function containing local multipliers and a nonsmooth penalty function. Then we construct a distributed continuous-time algorithm by virtue of a projected primal-dual subgradient dynamics. Based on the nonsmooth analysis and Lyapunov function, we obtain the existence of the solution to the nonsmooth algorithm and its convergence.