Skip to main content

Advertisement

Log in

On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we consider a vector optimization problem involving approximately star-shaped functions. We formulate approximate vector variational inequalities in terms of Fréchet subdifferentials and solve the vector optimization problem. Under the assumptions of approximately straight functions, we establish necessary and sufficient conditions for a solution of approximate vector variational inequality to be an approximate efficient solution of the vector optimization problem. We also consider the corresponding weak versions of the approximate vector variational inequalities and establish various results for approximate weak efficient solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Jofré, A., Luc, D.T., Théra, M.: ε-subdifferential and ε-monotonicity. Nonlinear Anal. 33, 71–90 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ngai, H.V., Luc, D.T., Théra, M.: Approximate convex functions. J. Nonlinear Convex Anal. 1(2), 155–176 (2000)

    MathSciNet  MATH  Google Scholar 

  3. Ngai, H.V., Penot, J.-P.: Approximately convex functions and approximately monotone operators. Nonlinear Anal. 66, 547–564 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ngai, H.V., Penot, J.-P.: Semismoothness and directional subconvexity of functions. Pac. J. Optim. 3(2), 323–344 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Hiriart-Urruty, J.-B.: A short proof of the variational principle for approximate solutions of a minimization problem. Am. Math. Mon. 90(3), 206–207 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hiriart-Urruty, J.-B.: From convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimality. In: Clarke, F.H., Demyanov, V.F., Giannessi, F. (eds.) Nonsmooth Optimization and Related Topics. Ettore Majorana Internat. Sci. Ser. Phys. Sci., vol. 43, pp. 219–239. Plenum, New York (1989)

    Google Scholar 

  7. Jofré, A., Rockafellar, R.T., Wets, Roger, J.-B.: Variational inequalities and economic equilibrium. Math. Oper. Res. 32(1), 32–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hiriart-Urruty, J.-B.: Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Ponstein, J. (ed.) Convexity and Duality in Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 256, pp. 37–70 (1986)

    Chapter  Google Scholar 

  9. Jofré, A., Luc, D.T., Théra, M.: ε-Subdifferential calculus for nonconvex functions and ε-monotonicity. C. R. Acad. Sci. Paris Sér. I Math. 323(7), 735–740 (1996)

    MATH  Google Scholar 

  10. Ngai, H.V., Luc, D.T., Théra, M.: Extensions of fréchet ε-subdifferential calculus and applications. J. Math. Anal. Appl. 268(1), 266–290 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Amahroq, T., Penot, J.-P., Syam, A.: On the subdifferentiability of difference of two functions and local minimization. Set-Valued Anal. 16, 413–427 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Penot, J.-P.: Softness, sleekness and regularity properties in nonsmooth analysis. Nonlinear Anal. 68(9), 2750–2768 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Penot, J.-P.: The directional subdifferential of the difference of two convex functions. J. Glob. Optim. 49(3), 505–519 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Giannessi, F.: Theorems of 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, New York (1980)

    Google Scholar 

  15. Siddiqi, A.H., Ansari, Q.H., Ahmed, R.: On vector variational-like inequalities. Indian J. Pure Appl. Math. 28(8), 1009–1016 (1997)

    MathSciNet  MATH  Google Scholar 

  16. Giannessi, F.: On Minty variational principle. In: Giannessi, F., Komlósi, S., Tapcsáck, T. (eds.) New Trends in Mathematical Programming, pp. 93–99. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  17. Yang, X.Q., Yang, X.M.: Vector variational-like inequalities with pseudoinvexity. Optimization 55(1–2), 157–170 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Komlósi, S.: On the stampacchia and minty variational inequalities. In: Giorgi, G., Rossi, F. (eds.) Generalized Convexity and Optimization for Economic and Financial Decisions, pp. 231–260. Pitagora Editrice, Bologna (1999)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  20. Mishra, S.K., Noor, M.A.: On vector variational-like inequality problems. J. Math. Anal. Appl. 311(1), 69–75 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Mishra, S.K., Wang, S.Y., Lai, K.K.: On non-smooth α-invex functions and vector variational-like inequality. Optim. Lett. 2, 91–98 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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  MathSciNet  MATH  Google Scholar 

  25. Mishra, S.K., Laha, V., Verma, R.U.: Generalized vector variational-like inequalities and nonsmooth vector optimization of radially (η,α)-continuous functions. Adv. Nonlinear Var. Inequal. 14(2), 1–18 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Parida, J., Sahoo, M., Kumar, A.: A variational-like inequality problem. Bull. Aust. Math. Soc. 39, 225–231 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Noor, M.A.: Variational-like inequalities. Optimization 30, 323–330 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fang, Y.P., Huang, N.J.: Variational-like inequalities with generalized monotone mappings in Banach spaces. J. Optim. Theory Appl. 118, 327–338 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ansari, Q.H.: On generalized vector variational-like inequalities. Ann. Sci. Math. Qué. 19(2), 131–137 (1995)

    MathSciNet  MATH  Google Scholar 

  30. Ahmed, R., Husain, S.: Generalized multivalued vector variational-like inequalities. Adv. Nonlinear Var. Inequal. 4(1), 105–116 (2001)

    MathSciNet  Google Scholar 

  31. Khan, M.F., Salahuddin: On generalized vector variational-like inequalities. Nonlinear Anal. 59, 879–889 (2004)

    MathSciNet  MATH  Google Scholar 

  32. Ansari, Q.H.: A note on generalized vector variational-like inequalities. Optimization 41, 197–205 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lee, B.S., Lee, G.M., Kim, D.S.: Generalized vector variational-like inequalities on locally convex Housdorff topological vector spaces. Indian J. Pure Appl. Math. 28(1), 33–41 (1997)

    MathSciNet  MATH  Google Scholar 

  34. Mishra, S.K., Giorgi, G.: Invexity and Optimization. Springer, New York (2008)

    Book  MATH  Google Scholar 

  35. Mishra, S.K., Wang, S.-Y., Lai, K.K.: V-Invex Functions and Vector Optimization. Springer, New York (2008)

    Book  MATH  Google Scholar 

  36. Mishra, S.K., Wang, S.-Y., Lai, K.K.: Generalized Convexity and Vector Optimization. Springer, New York (2009)

    Google Scholar 

  37. Pareto, V.: Course d’Economie Politique. Rouge, Lausanne (1896)

    Google Scholar 

  38. Boţ, R.I., Nechita, D.-M.: On the Dini-Hadamard subdifferential of the difference of two functions. J. Glob. Optim. 50, 485–502 (2011)

    Article  MATH  Google Scholar 

  39. Mishra, S.K., Laha, V.: On directionally approximately star-shaped functions and vector variational inequalities. J. Optim. Theory Appl. (submitted for publication)

Download references

Acknowledgements

The research of the second author is supported by the Council of Scientific and Industrial Research, New Delhi, Ministry of Human Resources Development, Government of India Grant 20-06/2010 (i) EU-IV.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. K. Mishra.

Additional information

Communicated by Guang-ya Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mishra, S.K., Laha, V. On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities. J Optim Theory Appl 156, 278–293 (2013). https://doi.org/10.1007/s10957-012-0124-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0124-4

Keywords

Navigation