Robust Constrained Consensus and Inequality-Constrained Distributed Optimization With Guaranteed Differential Privacy and Accurate Convergence

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 <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula>-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 nonseparable. As a byproduct of the co-design, we also propose a new constrained consensus algorithm that can achieve rigorous <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula>-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.