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Nonsmooth variational-like inequalities and nonsmooth vector optimization

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Abstract

In this paper, we consider nonsmooth vector variational-like inequalities and nonsmooth vector optimization problems. By using the scalarization method, we define nonsmooth variational-like inequalities by means of Clarke generalized directional derivative and study their relations with the vector optimizations and the scalarized optimization problems. Some existence results for solutions of our nonsmooth variational-like inequalities are presented under densely pseudomonotonicity or pseudomonotonicity assumption.

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References

  1. Al-Homidan, S., Ansari, Q.H.: Generalized Minty vector variational-like inequalities and vector optimization problems. J. Optim. Theory Appl. 144, 1–11 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. 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: Recent Advances in Vector Optimization, 221–274. Springer, Berlin (2012)

  3. Ansari, Q.H., Lee, G.M.: Non-smooth vector optimization problems and Minty vector variational inequalities. J. Optim. Theory Appl. 145, 1–16 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ansari, Q.H., Rezaei, M., Zafarani, J.: Generalized vector variational-like inequalities and vector optimization. J. Global Optim. 53, 271–284 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ansari, Q.H., Yao, J.C.: A fixed point theorem and its applications to the system of variational inequalities. Bull. Austral. Math. Soc. 59, 433–442 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ansari, Q.H., Yao, J.C.: On nondifferentiable and nonconvex vector optimization problems. J. Optim. Theory Appl. 106, 487–500 (2000)

    Article  MathSciNet  Google Scholar 

  7. Ansari, Q.H., Yao, J.-C.: Recent Advances in Vector Optimization. Springer, Berlin (2012)

    Book  Google Scholar 

  8. Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York (2003)

  9. Crespi, G.P., Ginchev, I., Rocca, M.: Some remarks on the Minty vector variational principle. J. Math. Anal. Appl. 345, 165–175 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gang, X., Liu, S.: On Minty vector variational-like inequality. Comput. Math. Appl. 56, 311–323 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Kluwer Academic Publishers, Dordrecht (2000)

  13. Guu, S.-M., Li, J.: Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets. Nonlinear Anal. 71, 2847–2855 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lalitha, C.S., Mehta, M.: Vector variational inequalities with cone-pseudomonotone bifunctions. Optimization 54, 327–338 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Luc, D.T.: Existence results for densely pseudomonotone variational inequalities. J. Math. Anal. Appl. 254, 291–308 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mishra, S.K., Wang, S.Y.: Vector variational-like inequalities and non-smooth vector optimization problems. Nonlinear Anal. 64, 1939–1945 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rezaie, M., Zafarani, J.: Vector optimization and variational-like inequalities. J. Global Optim. 43, 47–66 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A.: Relationships between vector variational-like inequality and optimization problems. Eur. J. Oper. Res. 157, 113–119 (2004)

    Article  MATH  Google Scholar 

  19. Santos, L.B., Rojas-Medar, M., Ruiz-Garzón, G., Rufián-Lizana, A.: Existence of weakly efficient solutions in nonsmooth vector optimization. Appl. Math. Comput. 200, 547–556 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Yang, X.M., Yang, X.Q.: Vector variational-like inequality with pseudoinvexity. Optimization 55, 157–170 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This research was done during the stay of second author at King Fahd University of Petroleum & Minerals, Dhahran Saudi Arabia. It was supported by a KFUPM funded project No. IN 101009.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    M. Alshahrani, Q. H. Ansari & S. Al-Homidan

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India

    Q. H. Ansari

Authors
  1. M. Alshahrani
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  2. Q. H. Ansari
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  3. S. Al-Homidan
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Corresponding author

Correspondence to Q. H. Ansari.

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Authors are grateful to the King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities to carry out this research.

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Alshahrani, M., Ansari, Q.H. & Al-Homidan, S. Nonsmooth variational-like inequalities and nonsmooth vector optimization. Optim Lett 8, 739–751 (2014). https://doi.org/10.1007/s11590-013-0614-9

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  • DOI: https://doi.org/10.1007/s11590-013-0614-9

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