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Sigma-subdifferential and its application to minimization problem

  • Hui HuangEmail author
  • Chunyou Sun


In this paper, we study \(\sigma \)-subdifferentials of \(\sigma \)-convex functions. Two equivalent conditions for \(\sigma \)-convexity are given. The formula for the \(\sigma \)-subdifferential of a sum of two functions is established. In terms of \(\sigma \)-subdifferential and Clarke’s normal cone, some Fermat’s rules for minimization problems are obtained.


\(\sigma \)-Convex function \(\sigma \)-Subdifferential Fermat’s rule Banach space 

Mathematics Subject Classification

49J52 90C26 



The authors are greatly indebted to the reviewers and the Editor for their valuable comments. Huang’s research was supported by the National Natural Science Foundation of China (Grant No. 11461080). Sun’s research was supported by the National Natural Science Foundation of China (Grant Nos. 11522109 and 11871169).


  1. 1.
    Alizadeh, M.H., Roohi, M.: Some results on pre-monotone operators. Bull. Iran. Math. Soc. 43(6), 2085–2097 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alizadeh, M.H., Hadjisavvas, N., Roohi, M.: Local boundedness properties for generalized monotone operators. J. Convex Anal. 19(1), 49–61 (2012)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Alizadeh, M.H.: On generalized convex functions and generalized subdifferential. Optim. Lett. (2018). Google Scholar
  4. 4.
    Borwein, J.M.: Fifty years of maximal monotonicity. Optim. Lett. 4(4), 473–490 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  6. 6.
    Huang, H.: Coderivative conditions for error bounds of \(\gamma \)-paraconvex multifunctions. Set-Valued Var. Anal. 20(4), 567–579 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Huang, H., He, M.: Weak sharp solutions of mixed variational inequalities in Banach spaces. Optim. Lett. 12(1), 287–299 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Iusem, A.N., Kassay, G., Sosa, W.: An existence result for equilibrium problems with some surjectivity consequences. J. Convex Anal. 16(4), 807–826 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Jofre, A., Luc, D.T., Thera, M.: \(\epsilon \)-subdifferential and \(\epsilon \)-monotonicity. Nonlinear Anal. 33(1), 71–90 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Luc, D.T., Ngai, H.V., Thera, M.: On \(\epsilon \)-monotonicity and \(\epsilon \)-convexity. In: Ioffe, A. (ed.) Calculus of Variations and Differential Equations (Haifa, 1998). Research Notes in Mathematics Series, vol. 410, pp. 82–100. Chapman & Hall, Boca Raton (1999)Google Scholar
  11. 11.
    Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham (2018)CrossRefzbMATHGoogle Scholar
  12. 12.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Volume I: Basic Theory. Springer, Berlin (2006)CrossRefGoogle Scholar
  13. 13.
    Mordukhovich, B.S., Mou, L.: Necessary conditions for nonsmooth optimization problems with operator constraints in metric spaces. J. Convex Anal. 16(3–4), 913–937 (2009)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Rockafellar, R.T.: Convex functions and dual extremum problems. Ph.D. Thesis, Hardvard University, Cambridge (1963)Google Scholar
  15. 15.
    Rolewicz, S.: On \(\gamma \)-paraconvex multifunctions. Math. Jpn. 24(3), 293–300 (1979)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Schirotzek, W.: Nonsmooth Analysis. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  17. 17.
    Zheng, X.Y., Ng, K.F.: The Fermat rule for multifunctions on Banach spaces. Math. Program. Ser. A. 104(1), 69–90 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsYunnan UniversityKunmingPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China

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