An Augmented Lagrangian distributed algorithm for an in-network optimal resource allocation problem

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This paper studies distributed solutions for an optimal resource allocation problem over networked systems with connected graph communication topologies. The problem setting consists of a group of agents in a network cooperatively meeting a demand by supplying a resource whose commitment incurs a cost on them. The objective in the optimal resource allocation problem is to obtain a commitment value for each agent such that the total cost, which is the sum of the costs of the agents, is minimized. In this paper we discuss how using ideas from Augmented Lagrangian method for convex optimization problems with affine constrains, we can arrive at a distributed solutions whose convergence guarantees holds for networks where the local costs are convex. We also show that if the local costs are all strongly convex and their gradients are globally Lipschitz then the convergence guarantees are exponential and the results can be extended to a special class of time-varying network interaction topologies. Simulations illustrate our results.