Distributed convex optimization of time-varying cost functions for double-integrator systems using nonsmooth algorithms

In this paper, nonsmooth algorithms are derived to solve a time-varying distributed convex optimization problem for continuous-time multi-agent systems with double-integrator dynamics. The objective is to minimize the sum of local time-varying cost functions, each of which is only known to an individual agent through local interactions. Here the optimal point is time varying. First, a centralized approach is introduced to solve the optimization problem with strongly-convex time-varying functions for single-integrator dynamics. Second, the problem is solved in a distributed manner for multi-agent systems with double-integrator dynamics. A nonsmooth algorithm is proposed where each agent relies only on its own position and the relative positions and velocities between itself and its neighbors. Hence communication is not necessary if the agents are equipped with sensing capabilities. The trade off is that a more restricted assumption is imposed on feasible cost functions. Third, an estimator-tracking nonsmooth algorithm is proposed where each agent generates an internal signal using information communicated from its neighbors. Then each agent minimizes the team cost function by tracking this internal signal. Here the assumption on feasible cost functions is relaxed.