# Randomized Constraints Consensus for Distributed Robust Mixed-Integer Programming

## TL;DR

This paper introduces a randomized distributed algorithm for solving uncertain mixed-integer convex programs across a network, ensuring high-confidence optimality despite unreliable communication and local uncertainties.

## Contribution

It presents a novel asynchronous, randomized distributed method for robust mixed-integer programming with convergence guarantees and practical implementation on multi-core systems.

## Key findings

- Algorithm achieves finite-time consensus on optimal solutions.
- High-confidence solutions are feasible and nearly optimal.
- Effective in uncertain, distributed environments like sensor networks.

## Abstract

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.