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Bulletin of the Iranian Mathematical Society

, Volume 44, Issue 5, pp 1283–1294 | Cite as

\(\varepsilon \)-Subdifferential as an Enlargement of the Subdifferential

  • Mahboubeh Rezaie
  • Zahra Sadat Mirsaney
Original Paper
  • 18 Downloads

Abstract

This work introduces a remarkable property of enlargements of maximal monotone operators. The basic tool in our analysis is a family of enlargements, introduced by Svaiter. Using the fact that the \(\varepsilon \)-subdifferential operator can be regarded as an enlargement of the subdifferential, a sufficient condition for some calculus rules in convex analysis can be provided. We give several corollaries about \(\varepsilon \)-subdifferential and extend one of them to arbitrary enlargement.

Keywords

Subdifferential \(\varepsilon \)-Subdifferential Enlargement \(\hbox {Weak}^{*}\)-lower semicontinuity 

Mathematics Subject Classification

Primary 47H05 Secondary 47H04 26E25 47N10 

Notes

Acknowledgements

The authors would like to thank the referees for valuable suggestions and remarks. We thank Professor R. S. Burachik who provided insight and expertise that greatly assisted the research.

References

  1. 1.
    Attouch, H., Brézis, H.: Duality for the sum of convex functions in general Banach spaces. In: Barroso, J.A. (ed.) Aspects of Mathematics and its Applications, pp. 125–133. Elsevier, Amsterdam (1986)CrossRefGoogle Scholar
  2. 2.
    Aubin, J.P.: Optima and Equilibria, An Introduction to Nonlinear Analysis, Graduate Texts in Mathematics. Springer, New York (1998)Google Scholar
  3. 3.
    Azé, D.: Duality for the sum of convex functions in general normed spaces. Arch. Math. 6(6), 554–561 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bauschke, H.H., Borwein, J.M., Wang, X.: Fitzpatrick functions and continuous linear monotone operators. SIAM J. Optim. 18(3), 789–809 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Borwein, J.: A note on \( \varepsilon \)-subgradients and maximal monotonicity. Pac. J. Math. 103(2), 307–314 (1982)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Borwein, J.M., Burachik, R.S., Yao, L.: Condition for zero duality gap in convex programming. J. Nonlinear Convex Anal. 15(1), 167–190 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)zbMATHGoogle Scholar
  8. 8.
    Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Burachik, R.S., Iusem, A.N., Svaiter, B.F.: Enlargements of maximal monotone operators with applications to variational inequalities. Set Valued Anal. 5(2), 159–180 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Burachik, R.S., Iusem, A.N.: Set-valued mappings and enlargements of monotone operators. Springer Optimization and its Applications. Springer, New York (2008)Google Scholar
  11. 11.
    Burachik, R.S., Svaiter, B.F.: \( \varepsilon \)-enlargements of maximal monotone operators in Banach spaces. Set Valued Var. Anal. 7(2), 117–132 (1999)MathSciNetCrossRefGoogle Scholar
  12. 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)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Burachik, R.S., Svaiter, B.F.: Operating enlargements of monotone operators: new connections with convex functions. Pac. J. Optim. 2(3), 425–445 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fenchel, W.: On conjugate convex functions. Can. J. Math. 1, 73–77 (1949)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hiriart-Urruty, J.B.: Subdifferential calculus using \(\varepsilon \)-subdifferentials. J. Funct. Anal. 118(1), 154–166 (1993)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hiriart-Urruty, J.B., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials. Nonlinear Anal. 24(12), 1727–1754 (1995)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jofre, A., Théra, M., Luc, D.T.: \( \varepsilon \)-subdifferentials and \( \varepsilon \)-monotonicity. Nonlinear Anal. 33(1), 71–90 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Martinez-Legaz, J.E., Théra, M., Luc, D.T.: \( \varepsilon \)-subdifferentials in terms of subdifferentials. Set Valued Anal. 4(4), 327–332 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    László, S.: \(\theta \)-monotone operators and \(\theta \)-convex function. Taiwan. J. Math. 16(2), 733–759 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lemarechal, C.: Extensions diverses des méthodes de gradient et applications. Thése d’etát, Université de Paris IX (1980)Google Scholar
  21. 21.
    Minkowski, H.: Theorie der konvexen Körper. insbesondere Begründung ihres ober Flächenbegriffs, Gesammelte Abhandlungen II. Teubner, Leipzig (1911)Google Scholar
  22. 22.
    Minty, G.J.: On the monotonicity of the gradient of a convex function. Pac. J. Math. 14, 243–247 (1964)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rockafellar, R.T.: Extension of Fenchel’s duality theorem. Duke Math. J. 33, 81–89 (1966)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rockafellar, R.T.: Level sets and continuity of conjugate convex functions. Trans. Am. Math. Soc. 123, 46–63 (1966)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Svaiter, B.F.: A family of enlargements of maximal monotone operators. Set Valued Anal. 8(4), 311–328 (2000)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wanka, G.: Convex Analysis. Chemnitz University of Technology, Chemnitz (2016)Google Scholar
  28. 28.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing, Singapore (2002)CrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of SciencesUniversity of IsfahanIsfahanIran

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