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A Rockafellar theorem for cyclically quasi-monotone maps: The regular non-vanishing case

Author

Listed:
  • 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)

Abstract

In this paper, we study the connection between cyclic quasi-monotonicity and quasi-convexity, focusing on whether every cyclically quasi-monotone (possibly multi-valued) map is included in the normal cone operator of a quasi-convex function, in analogy with Rockafellar's theorem for convex functions. We provide a positive answer for locally Lipschitz, non-vanishing maps in any dimension, as well as for general multi-maps in dimension 1. We further discuss connections to revealed preference theory in economics and to L optimal transport. Finally, we present explicit constructions and examples, highlighting the main challenges that arise in the general case.

Suggested Citation

  • Luigi de Pascale & Paul Pegon, 2026. "A Rockafellar theorem for cyclically quasi-monotone maps: The regular non-vanishing case," Post-Print hal-05160124, HAL.
    • Handle: RePEc:hal:journl:hal-05160124
    DOI: 10.1142/S0219199726500719
    Note: View the original document on HAL open archive server: https://inria.hal.science/hal-05160124v3
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