  We address differential privacy for fully distributed optimization subject to
a shared inequality constraint. By co-designing the distributed optimization
mechanism and the differential-privacy noise injection mechanism, we propose
the first distributed constrained optimization algorithm that can ensure both
provable convergence to a global optimal solution and rigorous
$\epsilon$-differential privacy, even when the number of iterations tends to
infinity. Our approach does not require the Lagrangian function to be strictly
convex/concave, and allows the global objective function to be non-separable.
As a byproduct of the co-design, we also propose a new constrained consensus
algorithm that can achieve rigorous $\epsilon$-differential privacy while
maintaining accurate convergence, which, to our knowledge, has not been
achieved before. Numerical simulation results on a demand response control
problem in smart grid confirm the effectiveness of the proposed approach.
