Hahn-Banach and Sandwich Theorems for Equivariant Vector Lattice-Valued Operators and Applications

Antonio Boccuto

doi:10.2478/tmmp-2020-0015

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Hahn-Banach and Sandwich Theorems for Equivariant Vector Lattice-Valued Operators and Applications

Tatra Mountains Mathematical Publications

Volume 76 (2020): Issue 1 (December 2020)

By: Antonio Boccuto

Open Access

|Nov 2020

Abstract

We prove Hahn-Banach, sandwich and extension theorems for vector lattice-valued operators, equivariant with respect to a given group G of homomorphisms. As applications and consequences, we present some Fenchel duality and separation theorems, a version of the Moreau-Rockafellar formula and some Farkas and Kuhn-Tucker-type optimization results. Finally, we prove that the obtained results are equivalent to the amenability of G.

References

[1] ALIPRANTIS, CH. D.—BURKINSHAW, O.: Locally Solid Riesz Spaces with Applications to Economics. Second edition. In: Math. Surveys and Monographs Vol. 105,American Mathematical Society, Providence, RI, 2003.10.1090/surv/105
Search in Google Scholar Back to article
[2] ASDRUBALI, F.—BALDINELLI, G.—BIANCHI, F. — COSTARELLI, D.—ROTILI, A.—SERACINI, M.—VINTI, G.: Detection of thermal bridges from thermographic images by means of image processing approximation algorithms, Appl. Math. Comput. 317 (2018), 160–171.10.1016/j.amc.2017.08.058
Search in Google Scholar Back to article
[3] BALBÁS, A.—BALBÁS, R. —MAYORAL, S.: Portfolio choice and optimal hedging with general risk functions: a simplex-like algorithm, European J. Oper. Res. 192 (2009), no. (2) 603–620.10.1016/j.ejor.2007.09.028
Search in Google Scholar Back to article
[4] BERGOUNIOUX, M.: Introduction au Traitement Mathématique des Images - Méthodes Déterministes. Springer-Verlag, Berlin, 2015.10.1007/978-3-662-46539-4
Search in Google Scholar Back to article
[5] BOCCUTO, A.: Riesz spaces, integration and sandwich theorems, Tatra Mt. Math. Publ. 3 (1993), 213–230.
Search in Google Scholar Back to article
[6] BOCCUTO, A.: Hahn-Banach-type theorems and applications to optimization for partially ordered vector space-valued invariant operators, Real Anal. Exchange 44 (2019), no. 2, 333–368.10.14321/realanalexch.44.2.0333
Search in Google Scholar Back to article
[7] BOCCUTO, A.—CANDELORO, D.: Sandwich theorems, extension principles and amenability, Atti Sem. Mat. Fis. Univ. Modena 42 (1994), 257–271.
Search in Google Scholar Back to article
[8] BOCCUTO, A.—CANDELORO, D.: Integral and ideals in Riesz spaces, Inform. Sci. 179 (2009), 2891–2902.10.1016/j.ins.2008.11.001
Search in Google Scholar Back to article
[9] BOCCUTO, A.—GERACE, I.—GIORGETTI, V.: A blind source separation technique for document restoration, SIAM J. Imaging Sciences 12 (2019), no. 2, 1135–1162.10.1137/18M1188793
Search in Google Scholar Back to article
[10] BOCCUTO, A.—SAMBUCINI, A. R.: The monotone integral with respect to Riesz space-valued capacities, Rend. Mat. (Roma) 16 (1996), 491–524.
Search in Google Scholar Back to article
[11] BUSKES, G.: The Hahn-Banach theorem surveyed, Dissertationes Math. 327 (1993), 1–35.
Search in Google Scholar Back to article
[12] CANDELORO, D.—MESIAR, R.—SAMBUCINI, A. R.: A special class of fuzzy measures: Choquet integral and applications, Fuzzy Sets Systems 355 (2019), 83–99.10.1016/j.fss.2018.04.008
Search in Google Scholar Back to article
[13] CHBANI, Z.—MAZGOURI, Z.—RIAHI, H.: From convergence of dynamical equilibrium systems to bilevel hierarchical Ky Fan minimax inequalities and applications, Minimax Theory Appl. 4 (2019), no. 2 231–270.
Search in Google Scholar Back to article
[14] CHOJNACKI, W.: Sur un théorème de Day, un théorème de Mazur-Orlicz et une généralisation de quelques théorèmes de Silverman, Colloq. Math. 50 (1986), 57–262.10.4064/cm-50-2-257-262
Search in Google Scholar Back to article
[15] CLUNI, F.—COSTARELLI, D.—MINOTTI, A. M.—VINTI, G.: Applications of Sampling Kantorovich Operators to Thermographic Images for Seismic Engineering,J. Comput. Anal. Appl. 19 (2015), no. 4, 602–617.
Search in Google Scholar Back to article
[16] CLUNI, F.—COSTARELLI, D.—MINOTTI, A. M.—VINTI, G.: Enhancement of thermographic images as tool for structural analysis in earthquake engineering, NDT & E International 70 (2015), no. 4, 60–72.10.1016/j.ndteint.2014.10.001
Search in Google Scholar Back to article
[17] COHEN, T. S. — M. WELLING, M.: Group equivariant convolutional networks. In: Proceedings of The 33rd International Conference on Machine Learning (ICML) Vol. 48 (2016), pp. 2990–2999.
Search in Google Scholar Back to article
[18] CONSTANTINESCU, C.—FILTER, W.—WEBER, K.: Advanced Integration Theory, Kluwer Acad. Publ., Dordrecht, 1998.10.1007/978-94-007-0852-5
Search in Google Scholar Back to article
[19] COSTARELLI, D.–SERACINI, M.—VINTI, G.: A segmentation procedure of the pervious area of the aorta artery from CT images without contrast medium, Math. Methods Appl. Sci. 43 (2020), no. 1, 114–133.10.1002/mma.5838
Search in Google Scholar Back to article
[20] COSTARELLI, D.—SERACINI, M.—VINTI, G.: Approximation problems for digital image processing and applications. In: Computational science and its applications-ICSA 2018, Part I. In: Lecture Notes in Comput. Sci. Vol. 10960. Springer, Cham, 2018. pp. 19–31.10.1007/978-3-319-95162-1_2
Search in Google Scholar Back to article
[21] EATON, M. L.: Group Invariance Applications in Statistics. In: NSF-CBMS Regional Conference Series in Probability and Statistics, Vol. 1. Institute of Mathematical Statistics, Hayward, CA, 1989.10.1214/cbms/1462061029
Search in Google Scholar Back to article
[22] ELSTER, K.—H.—NEHSE, R.: Konjugierte Operatoren und Subdifferentiale, Math. Operationsforsch. Statist. 6 (1975), no. 4, 641–657.10.1080/02331937508842279
Search in Google Scholar Back to article
[23] ELSTER, K.-H.—NEHSE, R.: Necessary and sufficient conditions for the order-completeness of partially ordered vector spaces, Math. Nachr. 81 (1978), 301–311.10.1002/mana.19780810116
Search in Google Scholar Back to article
[24] FARKAS, J.: Theorie der einfachen Ungleichungen, J. Reine Angew. Math. 124 (1902), 1–27.10.1515/crll.1902.124.1
Search in Google Scholar Back to article
[25] FREMLIN, D. H.: Measure Theory, Vol. 2: Broad Foundations. Corrected second printing of the 2001 original. Torres Fremlin, Colchester, 2003.
Search in Google Scholar Back to article
[26] GOODFELLOW, I.—BENGIO, Y.—COURVILLE, A.: Deep Learning. MIT Press, Cambridge, MA, 2016.
Search in Google Scholar Back to article
[27] GWINNER, J.: Resultsof Farkastype, Numer. Funct. Anal. Optim. 9 (1987), 471–520.10.1080/01630568708816244
Search in Google Scholar Back to article
[28] KAWASAKI, T.—TOYODA, M.—WATANABE, T.: The Hahn-Banach theorem and the separation theorem in a partially ordered vector space, J. Nonlinear Anal. Optim. 2 (2011), no. 1, 111–117.
Search in Google Scholar Back to article
[29] KONDOR, R.—TRIVEDI, S.: On the generalization of equivariance and convolution in neural networks to the action of compact groups, In: Proceedings of the 35th International Conference on Machine Learning (ICML 2018), Stockholm, Sweden, Vol. 80 (2018), pp. 2747–2755.
Search in Google Scholar Back to article
[30] KÖTHE, G.: Topological Vector Spaces I. Springer-Verlag, Berlin 1969.
Search in Google Scholar Back to article
[31] KUHN, H.—TUCKER, A. W.: Nonlinear programming. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability Vol. 1950.University of California Press, Berkeley and Los Angeles, 1951, pp. 481–492.
Search in Google Scholar Back to article
[32] KUSRAEV, A. G.—KUTATELADZE, S. S.: Subdifferentials: Theory and Applications. Kluwer Academic Publ., Dordrecht, 1995.10.1007/978-94-011-0265-0
Search in Google Scholar Back to article
[33] LUXEMBURG, W. A. J.—MASTERSON, J. J.: An extension of the concept of the order dual of a Riesz space, Canad. J. Math. 19 (1976), 488–498.10.4153/CJM-1967-041-6
Search in Google Scholar Back to article
[34] MANGASARIAN, O. L.: Nonlinear Programming. Soc. Industrial Appl. Math., New York, 1994.10.1137/1.9781611971255
Search in Google Scholar Back to article
[35] PATERSON, A. L. T.: Amenability. In: Mathematical Surveys and Monographs Vol. 29, American Mathematical Society, Providence, RI, 1988.10.1090/surv/029
Search in Google Scholar Back to article
[36] PFLUG, G. CH.—RŐMISCH, W.: Modeling, Measuring and Managing Risk. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.10.1142/6478
Search in Google Scholar Back to article
[37] RICCERI, B.: On a minimax theorem: an improvement, a new proof and an overview of its applications, Minimax Theory Appl. 2 (2017), no. 1, 99–152.
Search in Google Scholar Back to article
[38] RUSZCZYŃSKI, A.—SHAPIRO, A.: Optimality and duality in stochastic programming. Stochastic programming. In: Handbooks Oper. Res. Management Sci. Vol. 10. Elsevier Sci. B. V., Amsterdam, 2003, pp. 65–139.10.1016/S0927-0507(03)10002-3
Search in Google Scholar Back to article
[39] SAMBUCINI, A. R.: The Choquet integral with respect to fuzzy measures and applications, Math. Slovaca 67 (2017), no. 6, 1427–1450.10.1515/ms-2017-0049
Search in Google Scholar Back to article
[40] SCHMITT, L. M.: An equivariant version of the Hahn-Banach theorem, Houston J. Math. 18 (1992), no. 3, 429–447.
Search in Google Scholar Back to article
[41] SILVERMAN, R.: Invariant means and cones with vector interiors, Trans. Amer. Math. Soc. 88 (1958), no. 1 75–79.10.1090/S0002-9947-1958-0095414-6
Search in Google Scholar Back to article
[42] SILVERMAN, R.—YEN, T.: Addendum to: “Invariant means and cones with vector interiors”, Trans. Amer. Math. Soc. 88 (1958) no. 2, 327–330.10.1090/S0002-9947-1958-0095415-8
Search in Google Scholar Back to article
[43] SIMONS, S.: The Hahn-Banach-Lagrange theorem, Optimization 56 (2007), no. 1–2, 149–169.10.1080/02331930600819969
Search in Google Scholar Back to article
[44] SIMONS, S.: From Hahn-Banach to Monotonicity. Second Edition, Springer-Verlag, 2008.
Search in Google Scholar Back to article
[45] WRIGHT, J. D. M.: Paradoxical decompositions of the cube and injectivity, Bull. Lond. Math. Soc. 22 (1990), 18–24.10.1112/blms/22.1.18
Search in Google Scholar Back to article
[46] ZAANEN, A. C.: Riesz Spaces, II. North-Holland Publ. Co., Amsterdam, 1983.
Search in Google Scholar Back to article
[47] ZOWE, J.: A duality theorem for a convex programming problem in order complete vector lattices, J. Math. Anal. Appl. 50 (1975), 273–287.10.1016/0022-247X(75)90022-0
Search in Google Scholar Back to article
[48] ZOWE, J.: The saddle point theorem of Kuhn and Tucker in ordered vector spaces, J. Math. Anal. Appl. 57 (1977), 41–55.10.1016/0022-247X(77)90283-9
Search in Google Scholar Back to article
[49] ZOWE, J.: Sandwich theorems for convex operators with values in an ordered vector space, J. Math. Anal. Appl. 66 (1978), 282–296.10.1016/0022-247X(78)90232-9
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2020-0015 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

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Language: English

Page range: 11 - 34

Submitted on: Feb 18, 2020

Published on: Nov 4, 2020

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Fenchel duality theorem,

subgradient,

subdifferential,

Moreau-Rockafellar formula,

Related subjects:

© 2020 Antonio Boccuto, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 76 (2020): Issue 1 (December 2020)