Decentralized Laplacian Gradient Flows With Vanishing Anchors for Resource Allocation Problems With Arbitrary Initialization

Resource allocation problems are ubiquitous across different engineering applications where a fixed and limited asset needs to be optimally allocated among multiple agents in an online manner. While a plethora of algorithms exist for the solution of such problems, most existing approaches are restricted to either centralized initializations or the use of multiple auxiliary states that increase the complexity of the dynamics. In this letter, we tackle these challenges by introducing a novel and simple approach for the solution of decentralized resource allocation problems in multi-agent networked systems. The approach relies on a family of time-varying Laplacian gradient flows with vanishing anchors that remove any restriction on the initialization of the dynamics, without the need of incorporating additional auxiliary states. The convergence properties of the proposed dynamics are studied for a general class of time-varying vanishing gains. Analytical and numerical examples are also presented.