Continuous-time Distributed Convex Optimization with Set Constraints

Abstract

We study a distributed convex optimization problem with set constraints. The objective function is a summation of strictly convex functions. Based on a multi-agent system formulation, we consider that each node is with continuous-time dynamics and can only access its local objective function. Meanwhile, each node is subject to a common convex set constraint. The nodes can exchange local information with their neighbor nodes. A distributed gradient-based control protocol is applied to each node. It is shown that when the nodes are connected as an undirected graph and the time-varying gains of the gradients satisfy a persistence condition, the states of all the nodes will converge to the unique optimal point subject to the set constraints. Numerical examples are provided to demonstrate the results.