Multi-marginal maximal monotonicity and convex analysis
Abstract
Monotonicity and convex analysis arise naturally in the framework of multi-marginal optimal transport theory. However, a comprehensive multi-marginal monotonicity and convex analysis theory is still missing. To this end we study extensions of classical monotone operator theory and convex analysis into the multi-marginal setting. We characterize multi-marginal c-monotonicity in terms of classical monotonicity and firmly nonexpansive mappings. We provide Minty type, continuity and conjugacy criteria for multi-marginal maximal monotonicity. We extend the partition of the identity into a sum of firmly nonexpansive mappings and Moreau's decomposition of the quadratic function into envelopes and proximal mappings into the multi-marginal settings. We illustrate our discussion with examples and provide applications for the determination of multi-marginal maximal monotonicity and multi-marginal conjugacy. We also point out several open questions.
Cite
@article{arxiv.1901.03777,
title = {Multi-marginal maximal monotonicity and convex analysis},
author = {Sedi Bartz and Heinz H. Bauschke and Hung M. Phan and Xianfu Wang},
journal= {arXiv preprint arXiv:1901.03777},
year = {2019}
}