Distributed Constrained Continuous-Time Optimization With Input and Interaction Constraints

As is well known, it is challenging to address the convergence for distributed constrained optimization problem, in particular when nonconvex constraints, nonuniform step-sizes (nonuniform gradient gains) and switching graphs are involved. In this paper, we study the distributed constrained optimization problem in the presence of five kinds of nonlinearities caused by nonconvex control input constraints, nonconvex interaction constraints, nonuniform step-sizes, nonuniform convex state constraints and switching graphs. Due to the coupling of these nonlinearities, the interaction balance between agents does not exist any more and the edge weights are equivalent to be multiplied with time-varying factors, which results in the invalidness of the existing approaches. To decouple the nonlinearities, our approach is to construct an equivalent time-varying system and introduce a chain approach so as to show that the maximum distance from the agent states to the intersection set of the convex constraint state sets with a disturbance-like term decreases as time evolves. By combining the chain approach and a contradiction approach, it is proved that the optimization problem can be solved even when the five kinds of nonlinearities coexist. Finally, numerical examples are given to illustrate the theoretical results.