Peer-reviewedCC BY 4.0Communications in Contemporary Mathematics
A Rockafellar theorem for cyclically quasi-monotone maps: The regular non-vanishing case
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 L᪲ optimal transport. Finally, we present explicit constructions and examples, highlighting the main challenges that arise in the general case.
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BibTeX
@article{pascale2026rockafellar,
title = {{A Rockafellar theorem for cyclically quasi-monotone maps: The regular non-vanishing case}},
author = {Luigi de Pascale and Paul Pegon},
journal = {Communications in Contemporary Mathematics},
year = {2026},
month = sep,
doi = {10.1142/s0219199726500719},
eprint = {2507.09437},
archivePrefix = {arXiv},
url = {https://inria.hal.science/hal-05160124v3},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Rockafellar theorem for cyclically quasi-monotone maps: The regular non-vanishing case
AU - Luigi de Pascale
AU - Paul Pegon
PY - 2026
DA - 2026/09/22
JO - Communications in Contemporary Mathematics
DO - 10.1142/s0219199726500719
UR - https://inria.hal.science/hal-05160124v3
AB - 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 L᪲ optimal transport. Finally, we present explicit constructions and examples, highlighting the main challenges that arise in the general case.
SN - 0219-1997
SN - 0219-1997
ER -