Symbolic Computation with Monotone Operators

Abstract

We consider a class of monotone operators which are appropriate for symbolic representation and manipulation within a computer algebra system. Various structural properties of the class (e.g., closure under taking inverses, resolvents) are investigated as well as the role played by maximal monotonicity within the class. In particular, we show that there is a natural correspondence between our class of monotone operators and the subdifferentials of convex functions belonging to a class of convex functions deemed suitable for symbolic computation of Fenchel conjugates which were previously studied by Bauschke & von Mohrenschildt and by Borwein & Hamilton. A number of illustrative examples utilizing the introduced class of operators are provided including computation of proximity operators, recovery of a convex penalty function associated with the hard thresholding operator, and computation of superexpectations, superdistributions and superquantiles with specialization to risk measures.

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Acknowledgements

DRL was supported in part by Deutsche Forschungsgemeinschaft Collaborative Research Center SFB755. MKT was supported by Deutsche Forschungsgemeinschaft RTG2088.

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Correspondence to Matthew K. Tam.

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Dedicated to the memory of Jonathan Michael Borwein.

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Lauster, F., Luke, D.R. & Tam, M.K. Symbolic Computation with Monotone Operators. Set-Valued Var. Anal 26, 353–368 (2018). https://doi.org/10.1007/s11228-017-0418-7

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Keywords

  • Monotone operator
  • Symbolic computation
  • Experimental mathematics

Mathematics Subject Classification (2010)

  • 47H05
  • 47N10
  • 68W30