  In this paper, we deal with a network of agents that want to cooperatively
minimize the sum of local cost functions depending on a common decision
variable. We consider the challenging scenario in which objective functions are
unknown and agents have only access to local measurements of their local
functions. We propose a novel distributed algorithm that combines a recent
gradient tracking policy with an extremum-seeking technique to estimate the
global descent direction. The joint use of these two techniques results in a
distributed optimization scheme that provides arbitrarily accurate solution
estimates through the combination of Lyapunov and averaging analysis approaches
with consensus theory. We perform numerical simulations in a personalized
optimization framework to corroborate the theoretical results.
