Choquet representability of submodular functions

• 成果类型: Article
• 署名作者: Chateauneuf, Alain; Cornet, Bernard
• 署名单位: IPAG Business School; heSam Universite; Universite Pantheon-Sorbonne; Paris School of Economics; heSam Universite; Universite Pantheon-Sorbonne; University of Kansas
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-016-1074-7
• 发表日期: 2018
• 页码: 615-629
• 关键词: additivity

摘要

Let be an arbitrary set, equipped with an algebra and let be a functional defined on the set of bounded measurable functions . We provide necessary and sufficient conditions for a submodular functional f to be representable as a Choquet integral. From standard properties of the Choquet integral the functional f should be positively homogeneous and constant additive. Our first result shows that these two properties, together with submodularity, characterize a subadditive Choquet integral, when is finite. In the general case, f is a subadditive Choquet integral if and only if it satisfies the three previous properties, together with sup-norm continuity. This result provides another characterization of subadditive Choquet integrals different from the seminal paper by Schmeidler (Proc Am Math Soc 97(2):255-261, 1986) that relies on comonotonic additivity.