The quasi-convex Rockafellar theorem

Luigi de Pascale () and Paul Pegon ()
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Luigi de Pascale: DiMal - Dipartimento di Matematica e Informatica [Firenze] - UniFI - Università degli Studi di Firenze = University of Florence = Université de Florence
Paul Pegon: MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales - CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique - Centre Inria de Paris - Inria - Institut National de Recherche en Informatique et en Automatique, CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique

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Abstract: Rockafellar showed that every cyclically monotone operator is contained in the subdifferential of a convex function, and characterized the maximal ones. We establish the quasi-convex analogue of this theorem, with no assumption on the operator: in any separable Banach space, a setting we show to be sharp, every cyclically quasi-monotone operator is contained in the normal cone operator of a quasi-convex function. In finite dimension, we characterize maximal cyclically quasi-monotone operators and show that they are maximal quasi-monotone. Unlike in the convex case, the potential is not unique, and we build it from a preference relation obtained through successive preorders. As applications, we extend Afriat's theorem on revealed preferences in economics to infinite data sets, and obtain Kantorovich-type potentials for a supremal optimal transport problem.

Keywords: quasi-convex functions; cyclical quasi-monotonicity; quasi-monotone operators; revealed preference; L ∞ optimal transport; cyclical monotonicity; generalized convexity; Afriat's theorem; normal cone; utility representation (search for similar items in EconPapers)
Date: 2026-10-14
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