  We formalize the concept of Pareto Adaptive Robust Optimality (PARO) for
linear Adaptive Robust Optimization (ARO) problems. 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 such 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 particular approaches for finding and
approximating such solutions. 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.
