Afrika Matematika

, Volume 29, Issue 3–4, pp 383–398 | Cite as

On vector variational-like inequalities involving right upper-Dini-derivative functions

Article
  • 18 Downloads

Abstract

In this paper, we introduce (weak) Stampacchia and Minty arcwise connected vector variational-like inequalities in the term of right upper-Dini-derivative and establish not only the relations of introduced inequalities with vector optimization problems but also the existence results, by using KKM-Fan theorem and Brouwer fixed point theorem. Examples are provided to illustrate the derived results.

Keywords

Arcwise connected vector variational inequalities Vector optimization problems Arcwise connected functions Existence theorems 

Mathematics Subject Classification

90C30 90C29 49J40 

References

  1. 1.
    Al-Homidan, S., Ansari, Q.H.: Relations between generalized vector variational-like inequalities and vector optimization problems. Taiwan. J. Math. 16, 987–998 (2012)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Al-Homidan, S., Ansari, Q.H., Yao, J.-C.: Nonsmooth invexities, invariant monotonicities and nonsmooth vector variational-like inequalities with applications to vector optimization. In: Ansari, Q.H., Yao, J.-C. (eds.) In Recent Developments in Vector Optimization, pp. 221–274. Springer, Berlin (2012)CrossRefGoogle Scholar
  4. 4.
    Ansari, Q.H., Lalitha, C.S., Mehta, M.: Generalized Convexity, Nonsmooth Variational Inequalities and Nonsmooth Optimization. CRC Press, Taylor & Francis Group, Boca Raton (2014)MATHGoogle Scholar
  5. 5.
    Ansari, Q.H., Lee, G.M.: Nonsmooth vector optimization problems and Minty vector variational inequalities. J. Optim. Theory Appl. 145, 1–16 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ansari, Q.H., Rezaie, M., Zafarani, J.: Generalized vector variational-like inequalities and vector optimization. J. Glob. Optim. 53, 271–284 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Barbagallo, A., Pia, S.: Weighted variational inequalities in non-pivot Hilbert spaces with applications. Comput. Optim. Appl. 48, 487–514 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bhatia, D., Gupta, A., Arora, P.: Optimality via generalized approximate convexity and quasiefficiency. Optim. Lett. 7, 127–135 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ceng, L.C., Huang, S.: Existence theorems for generalized vector variational inequalities with a variable ordering relation. J. Glob. Optim. 46, 521–535 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Farajzadeh, A.P., Lee, B.S.: Vector variational-like inequality problem and vector optimization problem. Appl. Math. Lett. 23, 48–52 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ferreira, O.P.: Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds. Nonlinear Anal. 68, 1517–1528 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fu, J.Y., Wang, Y.H.: Arcwise connected cone-convex functions and mathematical programming. J. Optim. Theory Appl. 118, 339–352 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Giannessi, F.: Theorems of the alternative quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, Chichester (1980)Google Scholar
  14. 14.
    Giles, J.R.: Approximate Fréchet subdifferentability of convex functions. N. Z. J. Math. 37, 21–28 (2008)MATHGoogle Scholar
  15. 15.
    Gupta, A., Mehra, A., Bhatia, D.: Approximate convexity in vector optimization. Bull. Austral. Math. Soc. 74, 207–218 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hanson, M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lalitha, C.S., Mehta, M.: Vector variational inequalities with cone-pseudomonotone bifunctions. Optimization 54, 327–338 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Nishimura, H., Ok, E.A.: Solvability of variational inequalities on Hilbert lattices. Math. Oper. Res. 37, 608–625 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Noor, M.A., Noor, K.I., Yaqoob, H.: On general mixed variational inequalities. Acta Appl. Math. 110, 227–246 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Roshchina, V.: Mordukhovich subdifferential of pointwise minimum of approximate convex functions. Optim. Methods Softw. 25, 129–141 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Yang, X.M., Yang, X.Q.: Vector variational-like inequality with pseudoinvexity. Optimization 55, 157–170 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Yuan, D., Liu, X.: Mathematical programming involving \((\alpha,\rho )\)-right upper-Dini-derivative functions. Filomat 27, 899–908 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia

Personalised recommendations