  Real-time data-driven optimization and control problems over networks may
require sensitive information of participating users to calculate solutions and
decision variables, such as in traffic or energy systems. Adversaries with
access to coordination signals may potentially decode information on individual
users and put user privacy at risk. We develop local differential privacy,
which is a strong notion that guarantees user privacy regardless of any
auxiliary information an adversary may have, for a larger family of convex
distributed optimization problems. The mechanism allows agent to customize
their own privacy level based on local needs and parameter sensitivities. We
propose a general sampling based approach for determining sensitivity and
derive analytical bounds for specific quadratic problems. We analyze inherent
trade-offs between privacy and suboptimality and propose allocation schemes to
divide the maximum allowable noise, a privacy budget, among all participating
agents. Our algorithm is implemented to enable privacy in distributed optimal
power flow for electric grids.
