The quasi-convex Rockafellar theorem
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.
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BibTeX
@misc{pascale2026quasiconvex,
title = {{The quasi-convex Rockafellar theorem}},
author = {Luigi de Pascale and Paul Pegon},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20542},
eprint = {2610.20542},
archivePrefix = {arXiv},
url = {https://inria.hal.science/hal-05800300v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The quasi-convex Rockafellar theorem
AU - Luigi de Pascale
AU - Paul Pegon
PY - 2026
DA - 2026/10/14
PB - HAL
DO - 10.48550/arxiv.2610.20542
UR - https://inria.hal.science/hal-05800300v1
AB - 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.
ER -