Distributed optimization with the consideration of adaptivity and finite-time convergence

In this paper, a distributed optimization problem is studied for continuous-time multi-agent systems with single integrator dynamics. The objective is for multiple agents to cooperatively optimize a team performance function formed by a sum of convex local objective functions with only local interaction and information while explicitly taking into account adaptivity and finite-time convergence. The continuous-time algorithms have applications in motion coordination of multi-agent systems. First, a distributed algorithm with a signum function is introduced for a class of convex local objective functions. A corresponding condition is then given to guarantee that all agents reach a consensus in finite time while minimizing the team performance function. Second, an adaptive distributed algorithm is presented. It is shown that the interaction gain of each agent can be adaptively adjusted according to the variation of the gradients of the convex local objective functions, and the algorithm can deal with general differentiable convex local objective functions. Third, a distributed tracking algorithm combined with a distributed estimation algorithm is proposed for a class of convex local objective functions. It is shown that all agents reach a consensus while minimizing the team performance function in finite time. Numerical examples are included to illustrate the obtained theoretical results.