Distributed continuous-time gradient-based algorithm for constrained optimization

In this paper, we consider distributed algorithm based on a continuous-time multi-agent system to solve constrained optimization problem. The global optimization objective function is taken as the sum of agents' individual objective functions under a group of convex inequality function constraints. Because the local objective functions cannot be explicitly known by all the agents, the problem has to be solved in a distributed manner with the cooperation between agents. Here we propose a continuous-time distributed gradient dynamics based on the KKT condition and Lagrangian multiplier methods to solve the optimization problem. We show that all the agents asymptotically converge to the same optimal solution with the help of a constructed Lyapunov function and a LaSalle invariance principle of hybrid systems.