Abstract
Given two point to set operators, one of which is maximally monotone, we introduce a new distance in their graphs. This new concept reduces to the classical Bregman distance when both operators are the gradient of a convex function. We study the properties of this new distance and establish its continuity properties. We derive its formula for some particular cases, including the case in which both operators are linear monotone and continuous. We also characterize all bi-functions D for which there exists a convex function h such that D is the Bregman distance induced by h.
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Bauschke, H.H., Borwein, J.M., Wang, X.: Fitzpatrick function and continuous linear operators. SIAM J. Optim. 18(3), 789–809 (2007)
Borwein, J.M., Fitzpatrick, S., Vanderwerff, J.D.: Examples of convex functions and classifications of normed spaces. J. Convex Anal. 1(1), 61–73 (1994)
Borwein, J.M., Vanderwerff, J.D.: Convex functions: Constructions, characterizations and counterexamples. Cambridge University Press, Cambridge (2010)
Bueno, O., Martínez-Legaz, J.E., Svaiter, B.F.: On the monotone polar and representable closures of monotone operators. J. Convex Anal 21(2), 495–505 (2014)
Burachik, R.S., Fitzpatrick, S.: On the Fitzpatrick family associated to some subdifferentials. J. Nonlinear Convex Anal. 6(1), 165–171 (2005)
Burachik, R.S., Iusem, A.N.: Set valued mappings and enlargements of monotone operators. Springer Optimization and Its Applications, vol. 8. Springer, New York (2008)
Burachik, R.S., Iusem, A.N., Svaiter, B.F.: Enlargements of maximal monotone operators with application to variational inequalities. Set-Valued Anal. 5, 159–180 (1997)
Burachik, R., Martínez-Legaz, J.E., Rocco, M.: On a sufficient condition for equality of two maximal monotone operators. Set-Valued Variational Anal. 18(3-4), 327–335 (2010)
Burachik, R.S., Sagastizábal, C. A., Svaiter, B.F.: ε-enlargements of maximal monotone operators: theory and applications. In: Fukushima, M., Qi, L. (eds.) Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 25–43. Kluwer, Dordrecht (1997)
Burachik, R.S., Sagastizábal, C.A., Svaiter, B.F.: Bundle methods for maximal monotone operators. In: Tichatschke, R., Théra, M. (eds.) Ill–Posed Variational Problems and Regularization Techniques, pp. 49–64. Springer, Berlin (1999)
Burachik, R.S., Svaiter, B.F.: ε-enlargements of maximal monotone operators in Banach spaces. Set-Valued Anal. 7, 117–132 (1999)
Burachik, R.S., Svaiter, B.F.: Maximal monotone operators, convex functions and a special family of enlargements. Set-Valued Anal. 10(4), 297–316 (2002)
Burachik, R.S., Svaiter, B.F.: Enlargements of monotone operators: new connection with convex functions. Pacific Journal of Optimization 2(3), 425–445 (2006)
Diestel, J.: Sequences and series in banach spaces, graduate texts in mathematics, Springer-Verlag (1984)
Fitzpatrick, S.: Representing monotone operators by convex functions, pp. 59–65. Functional Analysis and Optimization, Workshop and Miniconference, Australia (1988). Proc. Center Math. Anal. Australian Nat. Univ. 20
Kiwiel, K.: Proximal minimization methods with generalized bregman functions. SIAM J. Control Optim. 35(4), 1142–1168 (1997)
Martínez-Legaz, J.E., Svaiter, B.F.: Monotone operators representable by l.s.c. convex functions. Set-Valued Anal. 13(1), 21–46 (2005)
Penot, J.P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58(7-8), 855–871 (2004)
Phelps, R.R.: Convex functions, monotone operators and differentiability. Springer, New York (2013)
Simons, S.: Minimax and monotonicity. Springer, Berlin (1998)
Simons, S.: Positive sets and monotone sets. J. Convex Anal. 14(2), 297–317 (2007)
Svaiter, B.F.: A family of enlargements of maximal monotone operators. Set-Valued Anal. 8, 311–328 (2000)
Acknowledgments
This author was partially supported by the MINECO of Spain, Grant MTM2014-59179- C2-2-P and by the Severo Ochoa Programme for Centres of Excellence in R&D [SEV-2015-0563]. He is affiliated with MOVE (Markets, Organizations and Votes in Economics) and with BGSMath (Barcelona Graduate School of Mathematics).
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Burachik, R.S., Martínez-Legaz, J.E. On Bregman-Type Distances for Convex Functions and Maximally Monotone Operators. Set-Valued Var. Anal 26, 369–384 (2018). https://doi.org/10.1007/s11228-017-0443-6
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DOI: https://doi.org/10.1007/s11228-017-0443-6
Keywords
- Maximally monotone operators
- Bregman distances
- Banach spaces
- Representable operators
- Fitzpatrick functions
- Convex functions
- Variational inequalities