  We study a class of cooperative multi-agent optimization problems, where each
agent is associated with a local action vector and a local cost, and the goal
is to cooperatively find the joint action profile that minimizes the average of
the local costs. Such problems arise in many applications, such as distributed
routing control, wind farm operation, etc. In many of these problems, gradient
information may not be readily available, and the agents may only observe their
local costs incurred by their actions as a feedback to determine their new
actions. In this paper, we propose a zeroth-order feedback optimization scheme
for the class of problems we consider, and provide explicit complexity bounds
for both the convex and nonconvex settings with noiseless and noisy local cost
observations. We also discuss briefly on the impacts of knowledge of local
function dependence between agents. The algorithm's performance is justified by
a numerical example of distributed routing control.
