Angle-Based Analysis Approach for Distributed Constrained Optimization

In this article, a distributed constrained optimization problem is studied with nonconvex input constraints, nonuniform convex state constraints, and nonuniform step sizes for single-integrator multiagent systems. Due to the existence of nonconvex input constraints, the edge weights between agents are equivalently multiplied with different time-varying scaling factors, and thus, the real interaction relationship cannot be kept balanced, even if the original communication graphs are kept balanced. Due to the existence of nonuniform convex state constraints and nonuniform step sizes, the system contains strong nonlinearities, which are coupled with the unbalance of the real interaction relationship, making existing analysis approaches hard to apply in this article. The main idea of the analysis approach is to fully explore the angles between the vectors from the agent states to their own projections on the intersection set of the convex state constraint sets so as to show that the distances from the agents to the intersection set diminish to zero as time evolves. By combining the analysis approaches in this article and our previous works, all agents are proved to converge to a common point and simultaneously solve the given optimization problem as long as the union of the communication graphs is strongly connected and balanced among each time interval of certain length. Numerical examples are given to show the obtained theoretical results.