Designing networks with resiliency to edge failures using two-stage robust optimization

We study the design of resilient single-commodity flow networks that can remain robust against multiple concurrent edge failures. We model these failures as binary random variables, allowing us to formally formulate the network design problem as a two-stage robust optimization problem. With an objective of minimizing the overall cost of building and operating the network, the capacities of the edges are decided in the first stage, while the optimal flows are determined in the second stage once the uncertainty has been realized. We first examine the standard affine decision rules approach and show that it is not a viable approach when two or more edges are allowed to fail at the same time. We then propose a column and constraint generation algorithm that we tailor to this application. Since the problem does not satisfy the relatively complete recourse assumption, we employ an oracle with two subproblems: one to determine edge failure scenarios that render the required demand satisfaction infeasible, and if no such scenario exists, a second one to determine the flow rerouting plan of highest cost. Our column and constraint generation algorithm is applied to networks adapted from the Survivable Network Design Library. For each instance, we determine sequences of fully adaptive, robust optimal solutions for various levels of resiliency, identifying also the maximum number of concurrent edge failures that can be sustained by these networks. Finally, we demonstrate how our algorithm can be applied to a defender versus attacker context, via the use of a decision-dependent uncertainty set.