  We provide a unifying framework for distributed convex optimization over
time-varying networks, in the presence of constraints and uncertainty, features
that are typically treated separately in the literature. We adopt a proximal
minimization perspective and show that this set-up allows us to bypass the
difficulties of existing algorithms while simplifying the underlying
mathematical analysis. We develop an iterative algorithm and show convergence
of the resulting scheme to some optimizer of the centralized problem. To deal
with the case where the agents' constraint sets are affected by a possibly
common uncertainty vector, we follow a scenario-based methodology and offer
probabilistic guarantees regarding the feasibility properties of the resulting
solution. To this end, we provide a distributed implementation of the scenario
approach, allowing agents to use a different set of uncertainty scenarios in
their local optimization programs. The efficacy of our algorithm is
demonstrated by means of a numerical example related to a regression problem
subject to regularization.
