  We study a class of distributed convex constrained optimization problems
where a group of agents aim to minimize the sum of individual objective
functions while each desires that any information about its objective function
is kept private. We prove the impossibility of achieving differential privacy
using strategies based on perturbing the inter-agent messages with noise when
the underlying noise-free dynamics are asymptotically stable. This justifies
our algorithmic solution based on the perturbation of individual functions with
Laplace noise. To this end, we establish a general framework for differentially
private handling of functional data. We further design post-processing steps
that ensure the perturbed functions regain the smoothness and convexity
properties of the original functions while preserving the differentially
private guarantees of the functional perturbation step. This methodology allows
us to use any distributed coordination algorithm to solve the optimization
problem on the noisy functions. Finally, we explicitly bound the magnitude of
the expected distance between the perturbed and true optimizers which leads to
an upper bound on the privacy-accuracy trade-off curve. Simulations illustrate
our results.
