  Many resource allocation problems can be formulated as an optimization
problem whose constraints contain sensitive information about participating
users. This paper concerns solving this kind of optimization problem in a
distributed manner while protecting the privacy of user information. Without
privacy considerations, existing distributed algorithms normally consist in a
central entity computing and broadcasting certain public coordination signals
to participating users. However, the coordination signals often depend on user
information, so that an adversary who has access to the coordination signals
can potentially decode information on individual users and put user privacy at
risk. We present a distributed optimization algorithm that preserves
differential privacy, which is a strong notion that guarantees user privacy
regardless of any auxiliary information an adversary may have. The algorithm
achieves privacy by perturbing the public signals with additive noise, whose
magnitude is determined by the sensitivity of the projection operation onto
user-specified constraints. By viewing the differentially private algorithm as
an implementation of stochastic gradient descent, we are able to derive a bound
for the suboptimality of the algorithm. We illustrate the implementation of our
algorithm via a case study of electric vehicle charging. Specifically, we
derive the sensitivity and present numerical simulations for the algorithm.
Through numerical simulations, we are able to investigate various aspects of
the algorithm when being used in practice, including the choice of step size,
number of iterations, and the trade-off between privacy level and
suboptimality.
