Constrained Shortest-Path Reformulations for Discrete Bilevel and Robust Optimization

Many discrete optimization problems are amenable to constrained shortest-path reformulations in an extended network space, a technique that has been key in convexification, bound strengthening, and search mechanisms for a large array of nonlinear problems. In this paper, we expand this methodology and propose constrained shortest-path models for challenging discrete variants of bilevel and robust optimization, where traditional methods (e.g., dualize-and-combine) are not applicable compared to their continuous counterparts. Specifically, we propose a framework that models problems as decision diagrams and introduces side constraints either as linear inequalities in the underlying polyhedral representation, or as state variables in shortest-path calculations. We show that the first case generalizes a common class of combinatorial bilevel problems where the follower's decisions are separable with respect to the leader's decisions. In this setting, the side constraints are indicator functions associated with arc flows, and we leverage polyhedral structure to derive an alternative single-level reformulation of the bilevel problem. The case where side constraints are incorporated as node states, in turn, generalizes classical robust optimization. For this scenario, we leverage network structure to derive an iterative augmenting state-space strategy akin to an L-shaped method. We evaluate both strategies on a bilevel competitive project selection problem and the robust traveling salesperson with time windows, observing considerable improvements in computational efficiency as compared to current state-of-the-art methods in the respective areas.