  In this paper, we introduce a new notion of guaranteed privacy that requires
that the change of the range of the corresponding inclusion function to the
true function is small. In particular, leveraging mixed-monotone inclusion
functions, we propose a privacy-preserving mechanism for nonconvex distributed
optimization, which is based on deterministic, but unknown, affine perturbation
of the local objective functions, which is stronger than probabilistic
differential privacy. The design requires a robust optimization method to
characterize the best accuracy that can be achieved by an optimal perturbation.
Subsequently, this is used to guide the refinement of a guaranteed-private
perturbation mechanism that can achieve a quantifiable accuracy via a
theoretical upper bound that is shown to be independent of the chosen
optimization algorithm.
