  In distributed optimization and iterative consensus literature, a standard
problem is for $N$ agents to minimize a function $f$ over a subset of Euclidean
space, where the cost function is expressed as a sum $\sum f_i$. In this paper,
we study the private distributed optimization (PDOP) problem with the
additional requirement that the cost function of the individual agents should
remain differentially private. The adversary attempts to infer information
about the private cost functions from the messages that the agents exchange.
Achieving differential privacy requires that any change of an individual's cost
function only results in unsubstantial changes in the statistics of the
messages. We propose a class of iterative algorithms for solving PDOP, which
achieves differential privacy and convergence to the optimal value. Our
analysis reveals the dependence of the achieved accuracy and the privacy levels
on the the parameters of the algorithm. We observe that to achieve
$\epsilon$-differential privacy the accuracy of the algorithm has the order of
$O(\frac{1}{\epsilon^2})$.
