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S-Convexity and Gross Substitutability.
- Source :
- Operations Research; May/Jun2024, Vol. 72 Issue 3, p1242-1254, 13p
- Publication Year :
- 2024
-
Abstract
- 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]
Details
- Language :
- English
- ISSN :
- 0030364X
- Volume :
- 72
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Operations Research
- Publication Type :
- Academic Journal
- Accession number :
- 177570059
- Full Text :
- https://doi.org/10.1287/opre.2022.2394