A Suite of Distributed Methodologies to Solve the Sparse Analytic Hierarchy Process Problem

In this paper we consider a scenario where a set of agents, interconnected by a network topology, aim at computing an estimate of their own utility, importance or value, based on pairwise relative information having heterogeneous nature. In more detail, the agents are able to measure the difference between their value and the value of some their neighbors, or have an estimate of the ratio between their value and the value the remaining neighbors. This setting may find application in problems involving information provided by heterogeneous sensors (e.g., differences and ratios), as well as in scenarios where estimations provided by humans have to be merged with sensor measurements. Specifically, we develop a distributed algorithm that lets each agent asymptotically compute a utility value. To this end, we first characterize the task at hand in terms of a least-squares minimum problem, providing a necessary and sufficient condition for the existence of a unique global minimum, and then we show that the proposed algorithm asymptotically converges to a global minimum. The paper is concluded by numerical analyses that corroborate the theoretical findings.