  This paper considers a time-varying optimization problem associated with a
network of systems, with each of the systems shared by (and affecting) a number
of individuals. The objective is to minimize cost functions associated with the
individuals' preferences, which are unknown, subject to time-varying
constraints that capture physical or operational limits of the network. To this
end, the paper develops a distributed online optimization algorithm with
concurrent learning of the cost functions. The cost functions are learned
on-the-fly based on the users' feedback (provided at irregular intervals) by
leveraging tools from shape-constrained Gaussian Processes. The online
algorithm is based on a primal-dual method, and acts effectively in a
closed-loop fashion where: i) users' feedback is utilized to estimate the cost,
and ii) measurements from the network are utilized in the algorithmic steps to
bypass the need for sensing of (unknown) exogenous inputs of the network. The
performance of the algorithm is analyzed in terms of dynamic network regret and
constraint violation. Numerical examples are presented in the context of
real-time optimization of distributed energy resources.
