Skip to main content
SpringerLink
Log in

Faces and Support Functions for the Values of Maximal Monotone Operators

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Representation formulas for faces and support functions of the values of maximal monotone operators are established in two cases: either the operators are defined on reflexive and locally uniformly convex real Banach spaces with locally uniformly convex duals, or their domains have nonempty interiors on real Banach spaces. Faces and support functions are characterized by the limit values of the minimal-norm selections of maximal monotone operators in the first case while in the second case they are represented by the limit values of any selection of maximal monotone operators. These obtained formulas are applied to study the structure of maximal monotone operators: the local unique determination from their minimal-norm selections, the local and global decompositions, and the unique determination on dense subsets of their domains.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer-Verlag, Berlin (1993)

    Book  Google Scholar 

  2. Hantoute, A., Nguyen, B.T.: Boundary of maximal monotone operators values. Appl. Math. Optim. 82, 225–243 (2020)

    Article  MathSciNet  Google Scholar 

  3. Papageorglou, N.S., Rădulescu, V.D., Repovš, D.D.: Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Cham (2019)

    Google Scholar 

  4. Borwein, J.M.: A note on \(\varepsilon \)-subgradients and maximal monotonicity. Pac. J. Math. 103, 307–314 (1982)

    Article  MathSciNet  Google Scholar 

  5. Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965)

    Article  MathSciNet  Google Scholar 

  6. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  Google Scholar 

  7. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (2009)

    Book  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics. Springer, New York (2010)

    Book  Google Scholar 

  11. Simons, S.: From Hahn-Banach to Monotonicity. Lecture Notes in Mathematics, vol. 1693, 2nd edn. Springer, New York (2008)

    MATH  Google Scholar 

  12. Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, vol. 1364, 2nd edn. Springer, Berlin (1993)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Phelps, R.R.: Lectures on maximal monotone operators. Extr. Math. 12, 193–230 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Borwein, J.M., Yao, L.: Structure theory for maximally monotone operators with points of continuity. J. Optim. Theory Appl. 157, 1–24 (2013)

    Article  MathSciNet  Google Scholar 

  16. Borwein, J.M., Yao, L.: Some results on the convexity of the closure of the domain of a maximally monotone operator. Optim. Lett. 8, 237–246 (2014)

    Article  MathSciNet  Google Scholar 

  17. Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)

    MATH  Google Scholar 

  18. Gasiński, L., Papageorgiou, N.S.: Exercises in Analysis, Part 2, Nonlinear Analysis. Springer, Cham (2016)

    Book  Google Scholar 

  19. Bishop, E., Phelps, R.R.: The support functionals of a convex set. Proc. Symp. Pure Math. 7, 27–35 (1963)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the editors and two anonymous referees for constructive comments and suggestions, which greatly improved the paper. Pham Duy Khanh was supported, in part, by the Fondecyt Postdoc Project 3180080, the Basal Program CMM–AFB 170001 from CONICYT–Chile, and the National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2017.325.

Author information

Authors and Affiliations

  1. Universidad de O’Higgins, Rancagua, Chile

    Bao Tran Nguyen

  2. Quy Nhon University, Quy Nhon, Vietnam

    Bao Tran Nguyen

  3. Department of Mathematics, HCMC University of Education, Ho Chi Minh, Vietnam

    Pham Duy Khanh

  4. Center for Mathematical Modeling, Universidad de Chile, Santiago, Chile

    Pham Duy Khanh

Authors
  1. Bao Tran Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pham Duy Khanh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pham Duy Khanh.

Additional information

Communicated by Constantin Zălinescu.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nguyen, B.T., Khanh, P.D. Faces and Support Functions for the Values of Maximal Monotone Operators. J Optim Theory Appl 186, 843–863 (2020). https://doi.org/10.1007/s10957-020-01737-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-020-01737-3

Keywords

Mathematics Subject Classification

Search

Navigation