On the Applications of Nonsmooth Vector Optimization Problems to Solve Generalized Vector Variational Inequalities Using Convexificators

  • Balendu Bhooshan UpadhyayEmail author
  • Priyanka Mishra
  • Ram N. Mohapatra
  • Shashi Kant Mishra
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)


In this paper, we employ the characterization for an approximate convex function in terms of its convexificator to establish the relationships between the solutions of Stampacchia type vector variational inequality problems in terms of convexificator and quasi efficient solution of a nonsmooth vector optimization problems involving locally Lipschitz functions. We identify the vector critical points, the weak quasi efficient points and the solutions of the weak vector variational inequality problem under generalized approximate convexity assumptions. The results of the paper extend, unify and sharpen corresponding results in the literature. In particular, this work extends and generalizes earlier works by Giannessi [11], Upadhyay et al. [31], Osuna-Gomez et al. [30], to a wider class of functions, namely the nonsmooth approximate convex functions and its generalizations. Moreover, this work sharpens earlier work by Daniilidis and Georgiev [5] and Mishra and Upadhyay [23], to a more general class of subdifferentials known as convexificators.


49J15 58E17 58E35 


  1. 1.
    Ansari, Q.H., Lee, G.M.: Nonsmooth vector optimization problems and Minty vector variational inequalities. J. Optim. Theory Appl. 145, 1–16 (2010)Google Scholar
  2. 2.
    Al-Homidan, S., Ansari, Q.H.: Generalized Minty vector variational like inequalities and vector optimization problems. J. Optim. Theory Appl. 144, 1–11 (2010)Google Scholar
  3. 3.
    Bhatia, D., Gupta, A., Arora, P.: Optimality via generalized approximate convexity and quasiefficiency. Optim. Lett. 7, 127–135 (2013)Google Scholar
  4. 4.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)Google Scholar
  5. 5.
    Daniilidis, A., Georgiev, P.: Approximate convexity and submonotonicity. J. Math. Anal. Appl. 291, 292–301 (2004)Google Scholar
  6. 6.
    Deng, S.: On approximate solutions in convex vector optimization. SIAM J. Control Optim. 35, 2128–2136 (1997)Google Scholar
  7. 7.
    Demyanov, V.F.: Convexification and Concavification of Positively Homogeneous Functions by the Same Family of Linear Functions. Report 3.208.802 Universita di Pisa (1994)Google Scholar
  8. 8.
    Demyanov, V.F., Jeyakumar, V.: Hunting for a smaller convex subdifferential. J. Global Optim. 10, 305–326 (1997)Google Scholar
  9. 9.
    Dutta, J., Chandra, S.: Convexificators, generalized convexity and vector optimization. Optimization 53, 77–94 (2004)Google Scholar
  10. 10.
    Dutta, J., Vetrivel, V.: On approximate minima in vector optimization. Numer. Funct. Anal. Optim. 22, 845–859 (2001)Google Scholar
  11. 11.
    Giannessi, F.: Theorems of the alternative, quadratic programming and complementarily problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, New York (1980)Google Scholar
  12. 12.
    Giannessi, F.: On Minty variational principle. In: Giannessi, F., Komlósi, S., Rapcsák, T. (eds.) New Trends in Mathematical Programming, pp. 93–99. Kluwer Academic Publishers, Dordrecht, Netherland (1997)Google Scholar
  13. 13.
    Golestani, M., Nobakhtian, S.: Convexificator and strong Kuhn-Tucker conditions. Comput. Math. Appl. 64, 550–557 (2012)Google Scholar
  14. 14.
    Gupta, A., Mehra, A., Bhatia, D.: Approximate convexity in vector optimization. Bull. Austral. Math. Soc. 74, 207–218 (2006)Google Scholar
  15. 15.
    Gupta, D., Mehra, A.: Two types of approximate saddle points. Numer. Funct. Anal. Optim. 29, 532–550 (2008)Google Scholar
  16. 16.
    Jeyakumar, V., Luc, D.T.: Nonsmooth calculus, minimality, and monotonicity of convexificators. J. Optim. Theory Appl. 101(3), 599–621 (1999)Google Scholar
  17. 17.
    Jeyakumar, V., Luc, D.T.: Approximate Jacobian matrices for nonsmooth continuous maps and \(C^1\)-optimization. SIAM J. Control Optim. 36, 1815–1832 (1998)Google Scholar
  18. 18.
    Li, X.F., Zhang, J.Z.: Stronger Kuhn-Tucker type conditions in nonsmooth multiobjective optimization: locally Lipschitz case. J. Optim. Theory Appl. 127, 367–388 (2005)Google Scholar
  19. 19.
    Long, X.J., Huang, N.J.: Optimality conditions for efficiency on nonsmooth multiobjective programming problems. Taiwanese J. Math. 18, 687–699 (2014)Google Scholar
  20. 20.
    Luu, D.V.: Convexifcators and necessary conditions for efficiency. Optimization 63, 321–335 (2013)Google Scholar
  21. 21.
    Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)Google Scholar
  22. 22.
    Michel, P., Penot, J.P.: A generalized derivative for calm and stable functions. Differ. Integral Equ. 5, 433–454 (1992)Google Scholar
  23. 23.
    Mishra, S.K., Upadhyay, B.B.: Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency. Positivity 17, 1071–1083 (2013)Google Scholar
  24. 24.
    Mishra, S.K., Upadhyay, B.B.: Pseudolinear Functions and Optimization. Taylor and Francis (2014)Google Scholar
  25. 25.
    Upadhyay, B.B., Mohapatra, R.N.: On approximate convex functions and submonotone operators using convexificators. J. Nonlinear Convex Anal. (2018) (submitted)Google Scholar
  26. 26.
    Mishra, S.K., Wang, S.Y., Lai, K.K.: Generalized Convexity and Vector Optimization. Nonconvex Optimization and Its Applications. Springer, Berlin (2009)Google Scholar
  27. 27.
    Mordukhovich, B.S., Shao, Y.H.: On nonconvex subdifferential calculus in Banach spaces. J. Convex Anal. 2, 211–227 (1995)Google Scholar
  28. 28.
    Ngai, H.V., Luc, D.T., Thera, M.: Approximate convex functions. J. Nonlinear Convex Anal. 1, 155–176 (2000)Google Scholar
  29. 29.
    Ngai, H.V., Penot, J.P.: Approximate convex functions and approximately monotone operators. Nonlinear Anal. 66, 547–564 (2007)Google Scholar
  30. 30.
    Osuna-Gomez, R., Rufian-Lizana, A., Ruiz-Canales, P.: Invex functions and generalized convexity in multiobjective programming. J. Optim. Theory Appl. 98, 651–661 (1998)Google Scholar
  31. 31.
    Upadhyay, B.B., Mohapatra, R.N., Mishra, S.K.: On relationships between vector variational inequality and nonsmooth vector optimization problems via strict minimizers. Adv. Nonlinear Var. Inequalities 20, 1–12 (2017)Google Scholar
  32. 32.
    Yang, X.M., Yang, X.Q., Teo, K.L.: Some remarks on the Minty vector variational inequality. J. Optim. Theory Appl. 121, 193–201 (1994)Google Scholar
  33. 33.
    Yang, X.Q., Zheng, X.Y.: Approximate solutions and optimality conditions of vector variational inequalities in Banach spaces. J. Glob. Optim. 40, 455–462 (2008)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology PatnaPatnaIndia
  2. 2.University of Central FloridaOrlandoUSA
  3. 3.Department of MathematicsInstitute of Science, Banaras Hindu UniversityVaranasiIndia

Personalised recommendations