Pareto Adaptive Robust Optimality via a Fourier–Motzkin Elimination lens

Abstract

We formalize the concept of Pareto Adaptive Robust Optimality (PARO) for linear two-stage Adaptive Robust Optimization (ARO) problems, with fixed continuous recourse. A worst-case optimal solution pair of here-and-now decisions and wait-and-see decisions is PARO if it cannot be Pareto dominated by another solution, i.e., there does not exist another worst-case optimal pair that performs at least as good in all scenarios in the uncertainty set and strictly better in at least one scenario. We argue that, unlike PARO, extant solution approaches—including those that adopt Pareto Robust Optimality from static robust optimization—could fail in ARO and yield solutions that can be Pareto dominated. The latter could lead to inefficiencies and suboptimal performance in practice. We prove the existence of PARO solutions, and present approaches for finding and approximating such solutions. Amongst others, we present a constraint & column generation method that produces a PARO solution for the considered two-stage ARO problems by iteratively improving upon a worst-case optimal solution. We present numerical results for a facility location problem that demonstrate the practical value of PARO solutions. Our analysis of PARO relies on an application of Fourier–Motzkin Elimination as a proof technique. We demonstrate how this technique can be valuable in the analysis of ARO problems, besides PARO. In particular, we employ it to devise more concise and more insightful proofs of known results on (worst-case) optimality of decision rule structures.