  This paper formalizes a demand response task as an optimization problem
featuring a known time-varying engineering cost and an unknown (dis)comfort
function. Based on this model, this paper develops a feedback-based projected
gradient method to solve the demand response problem in an online fashion,
where: i) feedback from the user is leveraged to learn the (dis)comfort
function concurrently with the execution of the algorithm; and, ii)
measurements of electrical quantities are used to estimate the gradient of the
known engineering cost. To learn the unknown function, a shape-constrained
Gaussian Process is leveraged; this approach allows one to obtain an estimated
function that is strongly convex and smooth. The performance of the online
algorithm is analyzed by using metrics such as the tracking error and the
dynamic regret. A numerical example is illustrated to corroborate the technical
findings.
