S-Convexity and Gross Substitutability

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.