  This paper considers distributed optimization (DO) where multiple agents
cooperate to minimize a global objective function, expressed as a sum of local
objectives, subject to some constraints. In DO, each agent iteratively solves a
local optimization model constructed by its own data and communicates some
information (e.g., a local solution) with its neighbors until a global solution
is obtained. Even though locally stored data are not shared with other agents,
it is still possible to reconstruct the data from the information communicated
among agents, which could limit the practical usage of DO in applications with
sensitive data. To address this issue, we propose a privacy-preserving DO
algorithm for constrained convex optimization models, which provides a
statistical guarantee of data privacy, known as differential privacy, and a
sequence of iterates that converges to an optimal solution in expectation. The
proposed algorithm generalizes a linearized alternating direction method of
multipliers by introducing a multiple local updates technique to reduce
communication costs and incorporating an objective perturbation method in the
local optimization models to compute and communicate randomized feasible local
solutions that cannot be utilized to reconstruct the local data, thus
preserving data privacy. Under the existence of convex constraints, we show
that, while both algorithms provide the same level of data privacy, the
objective perturbation used in the proposed algorithm can provide better
solutions than does the widely adopted output perturbation method that
randomizes the local solutions by adding some noise. We present the details of
privacy and convergence analyses and numerically demonstrate the effectiveness
of the proposed algorithm by applying it in two different applications, namely,
distributed control of power flow and federated learning, where data privacy is
of concern.
