Distributed Optimization With Nonconvex Velocity Constraints, Nonuniform Position Constraints, and Nonuniform Stepsizes

This note is devoted to the distributed optimization problem of multi-agent systems with nonconvex velocity constraints, nonuniform position constraints, and nonuniform stepsizes. Two distributed constrained algorithms with nonconvex velocity constraints and nonuniform stepsizes are proposed in the absence and the presence of nonuniform position constraints by introducing a switching mechanism to guarantee all agents' position states to remain in a bounded region. The algorithm gains need not to be predesigned and can be selected by each agent using its own and neighbors' information. By a model transformation, the original nonlinear time-varying system is converted into a linear time-varying one with a nonlinear error term. Based on the properties of stochastic matrices, it is shown that the optimization problem can be solved as long as the communication topologies are jointly strongly connected and balanced. One numerical example is given to show the obtained theoretical results.