  Convex optimization finds many real-life applications, where--optimized on
real data--optimization results may expose private data attributes (e.g.,
individual health records, commercial information), thus leading to privacy
breaches. To avoid these breaches and formally guarantee privacy to
optimization data owners, we develop a new privacy-preserving perturbation
strategy for convex optimization programs by combining stochastic
(chance-constrained) programming and differential privacy. Unlike standard
noise-additive strategies, which perturb either optimization data or
optimization results, we express the optimization variables as functions of the
random perturbation using linear decision rules; we then optimize these rules
to accommodate the perturbation within the problem's feasible region by
enforcing chance constraints. This way, the perturbation is feasible and makes
different, yet adjacent in the sense of a given distance function, optimization
datasets statistically similar in randomized optimization results, thereby
enabling probabilistic differential privacy guarantees. The chance-constrained
optimization additionally internalizes the conditional value-at-risk measure to
model the tolerance towards the worst-case realizations of the optimality loss
w.r.t. the non-private solution. We demonstrate the privacy properties of our
perturbation strategy analytically and through optimization and machine
learning applications.
