A differentially private distributed optimization method for constrained optimization

Abstract

We study a convex optimization problem with coupled inequality constraints over a multi-agent network while protecting the privacy of agents. Using the dual decomposition method, we first design a fully distributed dual algorithm. In the process of information exchange, dual (price) variables are injected artificially with the Laplace noise to preserve the privacy of local objective functions and constraint functions held by individual agents. Then we show that our algorithm can preserve the $ \epsilon $-differentially privacy by choosing an appropriate parameter of the Laplace noise. We also prove the convergence of the algorithm to the optimal solutions of primal and dual variables under the assumption of the decaying step-size. In addition, the trade-off between the level of privacy and the precision of the algorithm is discussed. Finally, several numerical studies are performed to validate the efficacy of the proposed algorithm.