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Representative functions of maximally monotone operators and bifunctions

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Abstract

The aim of this paper is to show that every representative function of a maximally monotone operator is the Fitzpatrick transform of a bifunction corresponding to the operator. In fact, for each representative function \(\varphi \) of the operator, there is a family of equivalent saddle functions (i.e., bifunctions which are concave in the first and convex in the second argument) each of which has \(\varphi \) as Fitzpatrick transform. In the special case where \(\varphi \) is the Fitzpatrick function of the operator, the family of equivalent saddle functions is explicitly constructed. In this way we exhibit the relation between the recent theory of representative functions, and the much older theory of saddle functions initiated by Rockafellar.

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References

  1. Alizadeh, M.H., Hadjisavvas, N.: Local boundedness of monotone bifunctions. J. Global Optim. 53, 231–241 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alizadeh, M.H., Hadjisavvas, N.: On the Fitzpatrick transform of a monotone bifunction. Optimization 62, 693–701 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13, 561–586 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Bot, R.I., Grad, S.-M.: Approaching the maximal monotonicity of bifunctions via representative functions. J. Convex Anal. 19, 713–724 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Burachik, R.S., Svaiter, B.F.: Maximal monotone operators, convex functions, and a special family of enlargements. Set Valued Anal. 10, 297–316 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fitzpatrick, S.: Representing monotone operators by convex functions. In: Workshop/Miniconference on Functional Analysis and Optimization (Canberra 1988) pp. 59–65, Procedings of the Centre for Mathematical Analysis, Australian National University, vol. 20. Australian National University, Canberra (1988)

  7. Hadjisavvas, N., Khatibzadeh, H.: Maximal monotonicity of bifunctions. Optimization 59, 147–160 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hadjisavvas, N., Jacinto, F.M.O., Martinez-Legaz, J.E.: Some conditions for maximal monotonicity of bifunctions. Set Valued Var. Anal. 24, 323–332 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Iusem, A.N.: On the maximal monotonicity of diagonal subdifferential operators. J. Convex Anal. 18, 489–503 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Krauss, E.: A representation of maximal monotone operators by saddle functions. Rev. Roumaine Math. Pures Appl. 30, 823–837 (1985)

    MathSciNet  MATH  Google Scholar 

  11. Krauss, E.: A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions. Nonlinear Anal. Theory Methods Appl. 9, 1381–1399 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  12. Marques Alves, M., Svaiter, B.F.: A new qualification condition for the maximality of the sum of maximal monotone operators in general Banach spaces. J. Convex Anal. 19, 575–589 (2012)

  13. Martinez-Legaz, J.-E., Svaiter, B.F.: Monotone operators representable by lsc convex functions. Set Valued Anal. 13, 21–46 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rockafellar, R.T.: Level sets and continuity of conjugate convex functions. Trans. Am. Math. Soc. 123, 46–61 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rockafellar, R.T.: Local boundedness of nonlinear monotone operators. Mich. Math. J. 16, 397–407 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rockafellar, R.T.: On the virtual convexity of the domain and range of a nonlinear maximal monotone operator. Math. Ann. 185, 81–90 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pacific J. Math. 33, 209–216 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rockafellar, R.T.: Convex Analysis. Princeton Mathematics, vol. 28. Princeton University Press, Princeton (1970)

    Book  Google Scholar 

  19. Rockafellar, R.T.: Saddle-points and convex analysis. In: Kuhn, H.W., Szego, G.P. (eds.) Differential Games and Related Topics, pp. 109–128. North-Holland Pub. co., Amsterdam (1971)

    Google Scholar 

  20. Simons, S.: Dualized and scaled Fitzpatrick functions. Proc. Am. Math. Soc. 134, 2983–2987 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Simons, S., Zalinescu, C.: Fenchel duality, Fitzpatrick functions and maximal monotonicity. J. Nonlinear Convex Anal. 6, 1–22 (2005)

    MathSciNet  MATH  Google Scholar 

  22. Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

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Acknowledgments

The authors would like to thank the referees for their suggestions that led to the improvement of the paper.

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Authors and Affiliations

  1. Università Cattolica del Sacro Cuore, Milan, Italy

    Monica Bianchi

  2. King Fahd University of Petroleum and Minerals, Dhahran, Kingdom of Saudi Arabia

    Nicolas Hadjisavvas

  3. Università degli Studi di Milano–Bicocca, Milan, Italy

    Rita Pini

Authors
  1. Monica Bianchi
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  2. Nicolas Hadjisavvas
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  3. Rita Pini
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Corresponding author

Correspondence to Rita Pini.

Additional information

Part of this work was done when Nicolas Hadjisavvas was visiting the Università Cattolica del Sacro Cuore, and the Università degli Studi di Milano–Bicocca, Italy. The author wishes to thank the Universities for their hospitality. Nicolas Hadjisavvas was supported by the startup research Grant No. SR141001 of the King Fahd University of Petroleum and Minerals.

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Bianchi, M., Hadjisavvas, N. & Pini, R. Representative functions of maximally monotone operators and bifunctions. Math. Program. 168, 433–448 (2018). https://doi.org/10.1007/s10107-016-1020-8

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  • DOI: https://doi.org/10.1007/s10107-016-1020-8

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