  This paper considers a distributed multi-agent optimization problem, with the
global objective consisting of the sum of local objective functions of the
agents. The agents solve the optimization problem using local computation and
communication between adjacent agents in the network. We present two randomized
iterative algorithms for distributed optimization. To improve privacy, our
algorithms add "structured" randomization to the information exchanged between
the agents. We prove deterministic correctness (in every execution) of the
proposed algorithms despite the information being perturbed by noise with
non-zero mean. We prove that a special case of a proposed algorithm (called
function sharing) preserves privacy of individual polynomial objective
functions under a suitable connectivity condition on the network topology.
