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Multi-marginal maximal monotonicity and convex analysis

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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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Acknowledgements

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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Authors and Affiliations

  1. Mathematics, University of Massachusetts Lowell, Lowell, MA, 01854, USA

    Sedi Bartz & Hung M. Phan

  2. Mathematics, University of British Columbia, Kelowna, BC, V1V 1V7, Canada

    Heinz H. Bauschke & Xianfu Wang

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  1. Sedi Bartz
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  2. Heinz H. Bauschke
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  3. Hung M. Phan
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  4. Xianfu Wang
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Correspondence to Sedi Bartz.

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Bartz, S., Bauschke, H.H., Phan, H.M. et al. Multi-marginal maximal monotonicity and convex analysis. Math. Program. 185, 385–408 (2021). https://doi.org/10.1007/s10107-019-01433-9

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  • DOI: https://doi.org/10.1007/s10107-019-01433-9

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