# Computational Methods for Path-based Robust Flows

## TL;DR

This paper introduces a computational approach for solving path-based robust flow problems under uncertainty, and proposes a new probabilistic model bridging stochastic and robust optimization, with theoretical and practical validation.

## Contribution

It presents a novel computational method for optimal robust flows and introduces a new probabilistic model that balances stochastic and robust optimization, applicable to general linear problems.

## Key findings

- The computational approach effectively solves robust flow problems to optimality.
- The new probabilistic model offers a practical compromise between stochastic and robust optimization.
- The approach extends to broader linear optimization problems.

## Abstract

Real world networks are often subject to severe uncertainties which need to be addressed by any reliable prescriptive model. In the context of the maximum flow problem subject to arc failure, robust models have gained particular attention. For a path-based model, the resulting optimization problem is assumed to be difficult in the literature, yet the complexity status is widely unknown. We present a computational approach to solve the robust flow problem to optimality by simultaneous primal and dual separation, the practical efficacy of which is shown by a computational study. Furthermore, we introduce a novel model of robust flows which provides a compromise between stochastic and robust optimization by assigning probabilities to groups of scenarios. The new model can be solved by the same computational techniques as the robust model. A bound on the generalization error is proven for the case that the probabilities are determined empirically. The suggested model as well as the computational approach extend to linear optimization problems more general than robust flows.