Set-Valued and Variational Analysis

, Volume 19, Issue 1, pp 75–96 | Cite as

Characterizing Convexity of a Function by Its Fréchet and Limiting Second-Order Subdifferentials

  • N. H. Chieu
  • T. D. Chuong
  • J.-C. YaoEmail author
  • N. D. Yen


The Fréchet and limiting second-order subdifferentials of a proper lower semicontinuous convex function \(\varphi: \mathbb R^n\rightarrow\bar{\mathbb R}\) have a property called the positive semi-definiteness (PSD)—in analogy with the notion of positive semi-definiteness of symmetric real matrices. In general, the PSD is insufficient for ensuring the convexity of an arbitrary lower semicontinuous function φ. However, if φ is a C 1,1 function then the PSD property of one of the second-order subdifferentials is a complete characterization of the convexity of φ. The same assertion is valid for C 1 functions of one variable. The limiting second-order subdifferential can recognize the convexity/nonconvexity of piecewise linear functions and of separable piecewise C 2 functions, while its Fréchet counterpart cannot.


Convexity Characterization Positive semi-definite property Fréchet second-order subdifferential Limiting second-order subdifferential 

Mathematics Subject Classifications (2000)

49K40 49J40 49J52 49J53 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bednarik, D., Pastor, K.: On characterizations of convexity for regularly locally Lipschitz functions. Nonlinear Anal. 57, 85–97 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)zbMATHGoogle Scholar
  3. 3.
    Cohn, D.L.: Measure Theory. Birkhäuser, Boston (1980)zbMATHGoogle Scholar
  4. 4.
    Cominetti, R., Correa, R.: A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim. 28, 789–809 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ginchev, I., Ivanov, V.I.: Second-order characterizations of convex and pseudoconvex functions. J. Appl. Anal. 9, 261–273 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.): Handbook of Generalized Convexity and Generalized Monotonicity. Springer, New York (2005)zbMATHGoogle Scholar
  8. 8.
    Hiriart-Urruty, J.-B., Strodiot, J.-J., Nguyen, V.H.: Generalized Hessian matrix and second-order optimality conditions for problems with C 1,1 data. Appl. Math. Optim. 11, 43–56 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jeyakumar, V., Yang, X.Q.: Approximate generalized Hessians and Taylor’s expansions for continuously Gâteaux differentiable functions. Nonlinear Anal. 36, 353–368 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Huang, L.R., Ng, K.F.: On lower bounds of the second-order directional derivatives of Ben-Tal, Zowe, and Chaney. Math. Oper. Res. 22, 747–753 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lee, G.M., Kim, D.S., Lee, B.S., Yen, N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. 34, 745–765 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: a Qualitative Study. Springer, New York (2005)zbMATHGoogle Scholar
  13. 13.
    Levy, A.B., Poliquin, R.A., Rockafellar, R.T.: Stability of locally optimal solutions. SIAM J. Optim. 10, 580–604 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field , D.A., Komkov, V. (eds.) Theoretical Aspects of Industrial Design. SIAM Proc. in Applied Mathematics, vol. 58, pp. 32–42. SIAM, Philadelphia (1992)Google Scholar
  15. 15.
    Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation. In: Basic Theory, vol. I. Applications, vol. II. Springer, Berlin (2006)Google Scholar
  17. 17.
    Mordukhovich, B.S., Outrata, J.V.: On second-order subdifferentials and their applications. SIAM J. Optim. 12, 139–169 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Phelps, R.R.: Convex functions, monotone operators and differentiability. In: Lecture Notes in Math, vol. 1364. Springer, Berlin (1993)Google Scholar
  19. 19.
    Poliquin, R.A., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8, 287–299 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  21. 21.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)zbMATHCrossRefGoogle Scholar
  22. 22.
    Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1974)zbMATHGoogle Scholar
  23. 23.
    Segal, I.E., Kunze, R.A.: Integrals and Operators. Springer, Berlin (1978)zbMATHGoogle Scholar
  24. 24.
    Vasilev, F.P.: Numerical Methods for Solving Extremal Problems, 2nd edn. Nauka, Moscow (In Russian) (1988)Google Scholar
  25. 25.
    Yang, X.Q.: Generalized second-order characterizations of convex functions. J. Optim. Theory Appl. 82, 173–180 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Yang, X.Q., Jeyakumar, V.: Generalized second-order directional derivatives and optimization with C 1,1 functions. Optimization 26, 165–185 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Yen, N.D.: Hölder continuity of solutions to a parametric variational inequality. Appl. Math. Optim. 31, 245–255 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Yen, N.D.: Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint. Math. Oper. Res. 20, 695–708 (1995)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • N. H. Chieu
    • 1
  • T. D. Chuong
    • 2
  • J.-C. Yao
    • 3
    Email author
  • N. D. Yen
    • 4
  1. 1.Department of MathematicsVinh UniversityVinhVietnam
  2. 2.Department of MathematicsDong Thap UniversityCao Lanh CityVietnam
  3. 3.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan
  4. 4.Institute of MathematicsVietnamese Academy of Science and TechnologyHanoiVietnam

Personalised recommendations