  In this article, we present an algorithm that drives the outputs of a network
of agents to jointly track the solutions of time-varying optimization problems
in a way that is robust to asynchrony in the agents' operations. We consider
three operations that can be asynchronous: (1) computations of control inputs,
(2) measurements of network outputs, and (3) communications of agents' inputs
and outputs. We first show that our algorithm converges to the solution of a
time-invariant feedback optimization problem in linear time. Next, we show that
our algorithm drives outputs to track the solution of time-varying feedback
optimization problems within a bounded error dependent upon the movement of the
minimizers and degree of asynchrony in a way that we make precise. These
convergence results are extended to quantify agents' asymptotic behavior as the
length of their time horizon approaches infinity. Then, to ensure satisfactory
network performance, we specify the timing of agents' operations relative to
changes in the objective function that ensure a desired error bound. Numerical
experiments confirm these developments and show the success of our distributed
feedback optimization algorithm under asynchrony.
