Preservation of supermodularity in parametric optimization: necessary and sufficient conditions on constraint structures

This paper presents a systematic study of the preservation of supermodularity under parametric optimization, allowing us to derive complementarity among parameters and monotonic structural properties for optimal policies in many operational models. We introduce the new concepts of mostly sublattice and additive mostly sublattice, which generalize the commonly imposed sublattice condition significantly, and use them to establish the necessary and sufficient conditions for the feasible set so that supermodularity can be preserved under various assumptions about the objective functions. Furthermore, we identify some classes of polyhedral sets that satisfy these concepts. Finally, we illustrate the use of our results in assemble-to-order systems.