Distributed Delay-Tolerant Strategies for Equality-Constraint Sum-Preserving Resource Allocation

Abstract

This paper proposes two nonlinear dynamics (one agent-based and one link-based) to solve a constrained distributed optimization (resource allocation) problem over a multi-agent network. The sum-preserving coupling refers to the constant amount of resources in the allocation problem. The proposed protocols consider model nonlinearities on agents,  including, but not limited to, quantization and saturation. Further, it can be adapted to specific applications, e.g.,  to reach faster convergence to the optimal value or to design robust dynamics in the presence of impulsive noise or uncertainties. The two nonlinear agent dynamics are considered in both continuous- and discrete-time. For both cases, we prove convergence over weakly connected networks using convex analysis and Lyapunov theory. The convergence proof, in general, is independent of the specific nonlinearities (or their composition) and holds for specific strongly sign-preserving nonlinear models. This work further proposes two different delay-tolerant mechanisms (in discrete-time) to handle bounded heterogeneous time-varying delays: (1) for general unknown delays, the protocol updates the states over a longer time scale so that at least one packet is received/sent over each link; (2) for known delays, the protocol updates at the same time-scale of the dynamics using all the received time-stamped information.We consider distributed generator power allocation over the smart grid as an application. This paper advances the state-of-the-art by addressing (i) possible nonlinearities on the agents' model, meanwhile handling (ii) solution feasibility at all times,  (iii) uniform-connectivity instead of all-time connectivity, and, finally, (iv) possible heterogeneous and time-varying delays over the network. To our best knowledge, no existing work addresses contributions (i)-(iv) altogether. Simulations are provided to evaluate and compare the performance over time-delayed and uniformly-connected networks for both quadratic and non-quadratic costs.

Keywords: Constrained distributed resource allocation, graph theory, convex analysis, time delay