  In this paper, we consider a network of processors aiming at cooperatively
solving mixed-integer convex programs subject to uncertainty. Each node only
knows a common cost function and its local uncertain constraint set. We propose
a randomized, distributed algorithm working under asynchronous, unreliable and
directed communication. The algorithm is based on a local computation and
communication paradigm. At each communication round, nodes perform two updates:
(i) a verification in which they check---in a randomized fashion---the robust
feasibility of a candidate optimal point, and (ii) an optimization step in
which they exchange their candidate basis (the minimal set of constraints
defining a solution) with neighbors and locally solve an optimization problem.
As main result, we show that processors can stop the algorithm after a finite
number of communication rounds (either because verification has been successful
for a sufficient number of rounds or because a given threshold has been
reached), so that candidate optimal solutions are consensual. The common
solution is proven to be---with high confidence---feasible and hence optimal
for the entire set of uncertainty except a subset having an arbitrary small
probability measure. We show the effectiveness of the proposed distributed
algorithm using two examples: a random, uncertain mixed-integer linear program
and a distributed localization in wireless sensor networks. The distributed
algorithm is implemented on a multi-core platform in which the nodes
communicate asynchronously.
