Distributed Continuous-Time Algorithms for Optimal Resource Allocation With Time-Varying Quadratic Cost Functions

In this article, we propose distributed continuous-time algorithms to solve the optimal resource allocation problem with certain time-varying quadratic cost functions for multiagent systems. The objective is to allocate a quantity of resources while optimizing the sum of all the local time-varying cost functions. Here, the optimal solutions are trajectories rather than some fixed points. We consider a large number of agents that are connected through a network, and our algorithms can be implemented using only local information. By making use of the prediction-correction method and the nonsmooth consensus idea, we first design two distributed algorithms to deal with the case when the time-varying cost functions have identical Hessians. We further propose an estimator-based algorithm which uses distributed average tracking theory to estimate certain global information. With the help of the estimated global information, the case of nonidentical constant Hessians is addressed. In each case, it is proved that the solutions of the proposed dynamical systems with certain initial conditions asymptotically converge to the optimal trajectories. We illustrate the effectiveness of the proposed distributed continuous-time optimal resource allocation algorithms through simulations.