Title: Randomized Constraints Consensus for Distributed Robust Linear Programming,
Abstract: In this paper we consider a network of processors aiming at cooperatively
solving linear programming problems 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 time-varying, asynchronous
and directed communication topology. 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
setup-the robust feasibility (and hence optimality) of the candidate optimal
point, and (ii) an optimization step in which they exchange their candidate
bases (minimal sets of active constraints) with neighbors and locally solve an
optimization problem whose constraint set includes: a sampled constraint
violating the candidate optimal point (if it exists), agent's current basis and
the collection of neighbor's basis. As main result, we show that if a processor
successfully performs the verification step for a sufficient number of
communication rounds, it can stop the algorithm since a consensus has been
reached. The common solution is-with high confidence-feasible (and hence
optimal) for the entire set of uncertainty except a subset having arbitrary
small probability measure. We show the effectiveness of the proposed
distributed algorithm on a multi-core platform in which the nodes communicate
asynchronously.