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On Bregman-Type Distances for Convex Functions and Maximally Monotone Operators

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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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References

  1. Bauschke, H.H., Borwein, J.M., Wang, X.: Fitzpatrick function and continuous linear operators. SIAM J. Optim. 18(3), 789–809 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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)

    MathSciNet  MATH  Google Scholar 

  3. Borwein, J.M., Vanderwerff, J.D.: Convex functions: Constructions, characterizations and counterexamples. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  4. 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)

    MathSciNet  MATH  Google Scholar 

  5. Burachik, R.S., Fitzpatrick, S.: On the Fitzpatrick family associated to some subdifferentials. J. Nonlinear Convex Anal. 6(1), 165–171 (2005)

    MathSciNet  MATH  Google Scholar 

  6. 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)

    Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

  10. 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)

  11. Burachik, R.S., Svaiter, B.F.: ε-enlargements of maximal monotone operators in Banach spaces. Set-Valued Anal. 7, 117–132 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Burachik, R.S., Svaiter, B.F.: Enlargements of monotone operators: new connection with convex functions. Pacific Journal of Optimization 2(3), 425–445 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Diestel, J.: Sequences and series in banach spaces, graduate texts in mathematics, Springer-Verlag (1984)

  15. 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

    MATH  Google Scholar 

  16. Kiwiel, K.: Proximal minimization methods with generalized bregman functions. SIAM J. Control Optim. 35(4), 1142–1168 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Martínez-Legaz, J.E., Svaiter, B.F.: Monotone operators representable by l.s.c. convex functions. Set-Valued Anal. 13(1), 21–46 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Penot, J.P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58(7-8), 855–871 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Phelps, R.R.: Convex functions, monotone operators and differentiability. Springer, New York (2013)

    MATH  Google Scholar 

  20. Simons, S.: Minimax and monotonicity. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  21. Simons, S.: Positive sets and monotone sets. J. Convex Anal. 14(2), 297–317 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Svaiter, B.F.: A family of enlargements of maximal monotone operators. Set-Valued Anal. 8, 311–328 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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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Authors and Affiliations

  1. School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes Campus SA, 5095, Adelaide, Australia

    Regina S. Burachik

  2. Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    Juan E. Martínez-Legaz

Authors
  1. Regina S. Burachik
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  2. Juan E. Martínez-Legaz
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Correspondence to Regina S. Burachik.

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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

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