Preservation of Supermodularity in Parametric Optimization: Necessary and Sufficient Conditions on Constraint Structures

The concept of supermodularity has received considerable attention in economics and operations research. It is closely related to the concept of complementarity in economics and has also proved to be an important tool for deriving monotonic comparative statics in parametric optimization problems and game theory models. However, only certain sufficient conditions (e.g., lattice structure) are identified in the literature to preserve the supermodularity. In this article, new concepts of mostly sublattice and additive mostly sublattice are introduced. With these new concepts, necessary and sufficient conditions for the constraint structures are established so that supermodularity can be preserved under various assumptions about the objective functions. Furthermore, some classes of polyhedral sets that satisfy these concepts are identified, and the results are applied to assemble-to-order systems. 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.