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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Bailey, D.H., Borwein, J.M.: Mathematics by Experiment: Plausible Reasoning in the 21st century. A K Peters Ltd, Natick, MA (2003)
Bailey, D.H., Borwein, J.M.: Experimental Mathematics: examples, methods and implications. Notices Amer. Math. Soc. 52(5), 502–514 (2005)
Bailey, D.H., Borwein, J.M., Calkin, N.J., Girgensohn, R., Luke, D.R., Moll, V.H.: Experimental Mathematics in Action. A K Peters Ltd, Natick (2007)
Bailey, D.H., Borwein, J.M., Girgensohn, R.: Experimentation in Mathematics: Computational Paths to Discovery. A K Peters Ltd, Natick (2003)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer Science & Business Media (2011)
Bauschke, H.H., von Mohrenschildt, M.: Fenchel conjugates and subdifferentials in Maple. Tech. rep. (1997)
Bauschke, H.H., von Mohrenschildt, M.: Symbolic computation of Fenchel conjugates. ACM Commun. Comput. Algebra 40(1), 18–28 (2006)
Bayram, I.: Penalty functions derived from monotone mappings. IEEE Signal Process. Lett. 22(3), 264–268 (2015)
Borwein, J.M., Hamilton, C.H.: Symbolic Fenchel conjugation. Math. Program. 116(1-2), 17–35 (2009). the accompanying SCAT Maple package is available online at http://vaopt.math.uni-goettingen.de/software.php
Lauster, F., Luke, D.R., Tam, M.K.: Maple worksheets for computational examples in Symbolic computation with monotone operators. Available online together with the SCAT package at http://vaopt.math.uni-goettingen.de/software.php
Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples, vol. 32. Cambridge University Press, Cambridge (2010)
Burachik, R., Iusem, A.N.: Set-valued mappings and enlargements of monotone operators, vol. 8. Springer Science & Business Media (2007)
Dentcheva, D., Martinez, G.: Two-stage stochastic optimization problems with stochastic ordering constraints on the recourse. Eur. J. Oper. Res 219, 1–8 (2012)
Gardiner, B., Lucet, Y.: Conjugate of convex piecewise linear-quadratic bivariate functions. Comput. Optim. Appl. 58, 249–272 (2014). doi:10.1007/s10589-013-9622-z
Hamilton, C.H.: Symbolic Convex Analysis. Master’s thesis, Simon Fraser University (2005)
Lucet, Y.: Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16(2), 171–185 (1997)
Lucet, Y.: What shape is your conjugate? A survey of computational convex analysis and its applications. SIAM Rev. 52(3), 505–542 (2010)
Niculescu, C., Persson, L.E.: Convex Functions and their Applications: A Contemporary Approach. Springer Science & Business Media (2006)
Ogryczak, W., Ruszczyński, A.: Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13, 60–78 (2002)
Rockafellar, R., Royset, J.: Random variables, monotone relations, and convex analysis. Math. Program. (2014) doi:10.1007/s10107-014-0801-1
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pacific J. Math. 33, 209–216 (1970)
Rockafellar, R.T., Wets, R.J.: Variational Analysis. Grundlehren Math. Wiss. Springer-Verlag, Berlin (1998)
Vakil, N.: Real Analysis Through Modern Infinitesimals. Encyclopedia of Mathematics and its Applications. Cambridge University Press, New York (2011)
DRL was supported in part by Deutsche Forschungsgemeinschaft Collaborative Research Center SFB755. MKT was supported by Deutsche Forschungsgemeinschaft RTG2088.
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
- Monotone operator
- Symbolic computation
- Experimental mathematics
Mathematics Subject Classification (2010)