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arXiv · 1901.03777

Multi-marginal maximal monotonicity and convex analysis

Sedi Bartz·Heinz H. Bauschke·Hung M. Phan·Xianfu Wang
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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.

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Sedi Bartz, Heinz H. Bauschke, Hung M. Phan, Xianfu Wang. 2019-01-12. Multi-marginal maximal monotonicity and convex analysis. https://arxiv.org/abs/1901.03777

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