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Hilbert direct integrals of monotone operators

Part of: Variational problems in infinite-dimensional spaces Existence theories Measures, integration, derivative, holomorphy

Published online by Cambridge University Press:  21 May 2024

Minh N. Bùi  and
Patrick L. Combettes
Minh N. Bùi
Affiliation:
Department of Mathematics and Scientific Computing, NAWI Graz, University of Graz, 8010 Graz, Austria e-mail: minh.bui@uni-graz.at
Patrick L. Combettes*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States
*
e-mail: plc@math.ncsu.edu
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Abstract

Finite Cartesian products of operators play a central role in monotone operator theory and its applications. Extending such products to arbitrary families of operators acting on different Hilbert spaces is an open problem, which we address by introducing the Hilbert direct integral of a family of monotone operators. The properties of this construct are studied, and conditions under which the direct integral inherits the properties of the factor operators are provided. The question of determining whether the Hilbert direct integral of a family of subdifferentials of convex functions is itself a subdifferential leads us to introducing the Hilbert direct integral of a family of functions. We establish explicit expressions for evaluating the Legendre conjugate, subdifferential, recession function, Moreau envelope, and proximity operator of such integrals. Next, we propose a duality framework for monotone inclusion problems involving integrals of linearly composed monotone operators and show its pertinence toward the development of numerical solution methods. Applications to inclusion and variational problems are discussed.

Keywords

Integration of set-valued mappingsmeasurable vector fieldmonotone operatoroptimizationvariational analysis

MSC classification

Primary: 49J53: Set-valued and variational analysis 46G10: Vector-valued measures and integration 58E30: Variational principles
Type
Article
Information
Canadian Journal of Mathematics , Volume 77 , Issue 4 , August 2025 , pp. 1425 - 1456
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Dedicated to the memory of Hédy Attouch

The work of P. L. Combettes was supported by the National Science Foundation under grant CCF-2211123.

References

Attouch, H., Familles d’opérateurs maximaux monotones et mesurabilité . Ann. Mat. Pura Appl. 120(1979), 35–111.CrossRefGoogle Scholar
Bartz, S., Bauschke, H. H., Borwein, J. M., Reich, S., and Wang, X., Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative . Nonlinear Anal. 66(2007), 1198–1223.CrossRefGoogle Scholar
Bauschke, H. H. and Combettes, P. L., Convex analysis and monotone operator theory in Hilbert spaces, 2nd ed., Springer, New York, 2017.CrossRefGoogle Scholar
Bismut, J.-M., Intégrales convexes et probabilités . J. Math. Anal. Appl. 42(1973), 639–673.CrossRefGoogle Scholar
Bogachev, V. I., Measure theory, Vol. I, Springer, Berlin, 2007.CrossRefGoogle Scholar
Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland/Elsevier, New York, 1973.Google Scholar
Browder, F. E., Nonlinear maximal monotone operators in Banach space . Math. Ann. 175(1968), 89–113.CrossRefGoogle Scholar
Browder, F. E. and Gupta, C. P., Monotone operators and nonlinear integral equations of Hammerstein type . Bull. Amer. Math. Soc. 75(1969), 1347–1353.CrossRefGoogle Scholar
Bùi, M. N. and Combettes, P. L., Warped proximal iterations for monotone inclusions . J. Math. Anal. Appl. 491(2020), Article no. 124315, 21 pp.CrossRefGoogle Scholar
Bùi, M. N. and Combettes, P. L., Multivariate monotone inclusions in saddle form . Math. Oper. Res. 47(2022), 1082–1109.CrossRefGoogle Scholar
Bùi, M. N. and Combettes, P. L., Interchange rules for integral functions. Preprint. https://arxiv.org/abs/2305.04872Google Scholar
Bùi, M. N. and Combettes, P. L., Integral resolvent and proximal mixtures. J. Optim. Theory Appl., to appear.Google Scholar
Butnariu, D. and Flåm, S. D., Strong convergence of expected-projection methods in Hilbert spaces . Numer. Funct. Anal. Optim. 16(1995), 601–636.CrossRefGoogle Scholar
Castaing, C. and Valadier, M., Convex analysis and measurable multifunctions, Springer, Berlin, 1977.CrossRefGoogle Scholar
Chaffey, T. and Sepulchre, R., Monotone one-port circuits . IEEE Trans. Autom. Control. 69(2024), 783–796.CrossRefGoogle Scholar
Combettes, P. L., The geometry of monotone operator splitting methods . Acta Numer. 33(2024), 487–632.CrossRefGoogle Scholar
Combettes, P. L. and Pesquet, J.-C., Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators . Set-Valued Var. Anal. 20(2012), 307–330.CrossRefGoogle Scholar
Combettes, P. L. and Woodstock, Z. C., Reconstruction of functions from prescribed proximal points . J. Approx. Theory 268(2021), Article no. 105606, 26 pp.CrossRefGoogle Scholar
Combettes, P. L. and Woodstock, Z. C., A variational inequality model for the construction of signals from inconsistent nonlinear equations . SIAM J. Imaging Sci. 15(2022), 84–109.CrossRefGoogle Scholar
Dixmier, J., Les algèbres d’opérateurs dans l’espace hilbertien (algèbres de von Neumann), 2e éd., Gauthier-Villars, Paris, 1969.Google Scholar
Ekeland, I. and Temam, R., Convex analysis and variational problems, SIAM, Philadelphia, PA, 1999.CrossRefGoogle Scholar
Ghoussoub, N., Self-dual partial differential systems and their variational principles, Springer, New York, 2009.Google Scholar
Godement, R., Théorie générale des sommes continues d’espaces de Banach . C. R. Acad. Sci. Paris. A228(1949), 1321–1323.Google Scholar
Godement, R., Sur la théorie des représentations unitaires . Ann. of Math. 53(1951), 68–124.CrossRefGoogle Scholar
Hytönen, T., van Neerven, J., Veraar, M., and Weis, L., Analysis in Banach spaces I – Martingales and Littlewood–Paley theory, Springer, New York, 2016.Google Scholar
Kondô, M., Sur les sommes directes des espaces linéaires . Proc. Imp. Acad. Tokyo 20(1944), 425–431.Google Scholar
Moreau, J. J., Propriétés des applications “prox. ” C. R. Acad. Sci. Paris A256(1963), 1069–1071.Google Scholar
Moreau, J. J., Proximité et dualité dans un espace hilbertien . Bull. Soc. Math. France 93(1965), 273–299.CrossRefGoogle Scholar
Nemirovski, A., Juditsky, A., Lan, G., and Shapiro, A., Robust stochastic approximation approach to stochastic programming . SIAM J. Optim. 19(2009), 1574–1609.CrossRefGoogle Scholar
von Neumann, J., On rings of operators. Reduction theory . Ann. of Math. 50(1949), 401–485 (written in 1938).CrossRefGoogle Scholar
Pennanen, T., Dualization of generalized equations of maximal monotone type . SIAM J. Optim. 10(2000), 809–835.CrossRefGoogle Scholar
Pennanen, T., Revalski, J. P., and Théra, M., Variational composition of a monotone operator and a linear mapping with applications to elliptic PDEs with singular coefficients . J. Funct. Anal. 198(2003), 84–105.CrossRefGoogle Scholar
Robinson, S. M., Composition duality and maximal monotonicity . Math. Program. 85(1999), 1–13.CrossRefGoogle Scholar
Rockafellar, R. T., Convex integral functionals and duality . In: Zarantonello, E. H. (ed.), Contributions to nonlinear functional analysis, Academic Press, New York, 1971, pp. 215–236.CrossRefGoogle Scholar
Rockafellar, R. T., Conjugate duality and optimization, SIAM, Philadelphia, PA, 1974.CrossRefGoogle Scholar
Schwartz, L., Analyse III – Calcul intégral, Hermann, Paris, 1993.Google Scholar
Showalter, R. E., Monotone operators in Banach space and nonlinear partial differential equations, American Mathematical Society, Providence, RI, 1997.Google Scholar
Zarantonello, E. H. (ed.), Contributions to nonlinear functional analysis, Academic Press, New York, 1971.Google Scholar