A dynamic programming approach for the decentralized control of discrete optimizers with quadratic utilities and shared constraint

This paper addresses the problem of controlling a large set of agents, each with a quadratic utility function depending on individual combinatorial choices, and all sharing an affine constraint on available resources. Such a problem is formulated as an integer mono-constrained bounded quadratic knapsack problem. Differently from the centralized approaches typically proposed in the related literature, we present a new decentralized algorithm to solve the problem approximately in polynomial time by decomposing it into a finite series of sub-problems. We assume a minimal communication structure through the presence of a central coordinator that ensures the information exchange between agents. The proposed solution relies on a decentralized control algorithm that combines discrete dynamic programming with additive decomposition and value functions approximation. The optimality and complexity of the presented strategy are discussed, highlighting that the algorithm constitutes a fully polynomial approximation scheme. Numerical experiments are presented to show the effectiveness of the approach in the optimal resolution of large-scale instances.