  We present an optimization framework for solving multi-agent nonlinear
programs subject to inequality constraints while keeping the agents' state
trajectories private. Each agent has an objective function depending only upon
its own state and the agents are collectively subject to global constraints.
The agents do not directly communicate with each other but instead route
messages through a trusted cloud computer. The cloud computer adds noise to
data being sent to the agents in accordance with the framework of differential
privacy in order to keep each agent's state trajectory private from all other
agents and any eavesdroppers. This private problem can be viewed as a
stochastic variational inequality and is solved using a projection-based method
for solving variational inequalities that resembles a noisy primal-dual
gradient algorithm. Convergence of the optimization algorithm in the presence
of noise is proven and a quantifiable trade-off between privacy and convergence
is extracted from this proof. Simulation results are provided that demonstrate
numerical convergence for both $\epsilon$-differential privacy and $(\epsilon,
\delta)$-differential privacy.
