S-Convexity and Gross Substitutability.

A New Concept to Study Substitute Structures in Economics and Operations Models In "S-Convexity and Gross Substitutability," Chen and Li propose a novel concept of S-convex functions defined on continuous spaces, which extends a key concept of M-natural-convex functions in discrete convex analysis. They develop a host of fundamental properties and characterizations of S-convex functions. In a parametric maximization model with a box constraint, they show that the set of optimal solutions is nonincreasing in the parameters if the objective function is S-concave and prove the necessity of S-concavity under some conditions. The monotonicity result finds notable inventory models. Interestingly, the authors show that S-concavity is the correct notion characterizing gross substitutability, an important concept in economics for markets with divisible goods. We propose a new concept of S-convex functions (and its variant, semistrictly quasi-S- (SSQS)-convex functions) to study substitute structures in economics and operations models with continuous variables. We develop a host of fundamental properties and characterizations of S-convex functions, including various preservation properties, conjugate relationships with submodular and convex functions, and characterizations using Hessians. For a divisible market, we show that the utility function satisfies gross substitutability if and only if it is S-concave under mild regularity conditions. In a parametric maximization model with a box constraint, we show that the set of optimal solutions is nonincreasing in the parameters if the objective function is (SSQS-) S-concave. Furthermore, we prove that S-convexity is necessary for the property of nonincreasing optimal solutions under some conditions. Our monotonicity result is applied to analyze two notable inventory models: a single-product inventory model with multiple unreliable suppliers and a classic multiproduct dynamic inventory model with lost sales. Funding: This work was supported by the National Science Foundation [Grants CMMI-1538451 and CMMI-1635160] and the gift fund from JD.com. Supplemental Material: The online companion is available at https://doi.org/10.1287/opre.2022.2394. [ABSTRACT FROM AUTHOR]