Multi-marginal maximal monotonicity and convex analysis
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.
Keywordsc-Convexity c-Monotonicity c-Splitting set Cyclic monotonicity Kantorovich duality Maximal monotonicity Minty theorem Moreau envelope Multi-marginal Optimal transport
Mathematics Subject ClassificationPrimary 47H05 26B25 Secondary 49N15 49K30 52A01 91B68
We thank three anonymous referees for their kind and useful remarks. Sedi Bartz was partially supported by a University of Massachusetts Lowell startup grant. Heinz Bauschke and Xianfu Wang were partially supported by the Natural Sciences and Engineering Research Council of Canada. Hung Phan was partially supported by Autodesk, Inc.
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