Nothing Special   »   [go: up one dir, main page]

Skip to main content
Log in

Optimality Conditions of Generalized Subconvexlike Set-Valued Optimization Problems Based on the Quasi-Relative Interior

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

Abstract

In this paper, firstly, a new generalized subconvexlike set-valued map based on the quasi-relative interior is introduced. Secondly, by a separation theorem involving the quasi-relative interior, some separation properties are obtained. Finally, some optimality conditions are established. Our results improve some results in the literature.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Borwein, J.M., Lewis, A.S.: Partially finite convex programming, Part I: quasi relative interiors and duality theory. Math. Program. 57(1–3), 15–48 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Jeyakumar, V., Wolkowicz, H.: Generalizations of Slater’s constraint qualification for infinite convex programs. Math. Program. 57(1–3), 85–101 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Limber, M.A., Goodrich, R.K.: Quasi interiors, Lagrange multipliers and l p spectral estimation with lattice bounds. J. Optim. Theory Appl. 78(1), 143–161 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Borwein, J.M., Goebel, R.: Notions of relative interior in Banach spaces. J. Math. Sci. 115, 2542–2553 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cammaroto, F., Di Bella, B.: Separation theorem based on the quasi relative interior and application to duality theory. J. Optim. Theory Appl. 125(1), 223–229 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cammaroto, F., Di Bella, B.: On a separation theorem involving the quasi relative interior. Proc. Edinb. Math. Soc. 50, 605–610 (2007)

    MATH  MathSciNet  Google Scholar 

  7. Daniele, P., Giuffré, S., Idone, G., Maugeri, A.: Infinite dimensional duality and applications. Math. Ann. 339(1), 221–239 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bot, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasi relative interior in convex programming. SIAM J. Optim. 19(1), 217–233 (2008)

    Article  MathSciNet  Google Scholar 

  9. Bot, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasi relative interior in convex optimization. J. Optim. Theory Appl. 139(1), 67–84 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bot, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization. doi:10.1080/02331934.2010.505649

  11. Bot, R.I., Grad, S.M., Wanka, G.: Duality in Vector Optimization. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  12. Grad, A.: Quasi relative interior-type constraint qualifications ensuring strong Lagrange duality for optimization problems with cone and affine constraints. J. Math. Anal. Appl. 361, 86–95 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Rong, W.D., Wu, Y.N.: Characterizations of super efficiency in cone-convexlike vector optimization with set-valued maps. Math. Methods Oper. Res. 48, 247–258 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Li, Z.M.: A theorem of the alternative and its Application to the optimization of set-valued maps. J. Optim. Theory Appl. 100(2), 365–375 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Yang, X.M., Yang, X.Q., Chen, G.Y.: Theorems of the alternative and optimization with set-valued maps. J. Optim. Theory Appl. 107(3), 627–640 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110(2), 413–427 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sach, P.H.: Nearly subconvexlike set-valued maps and vector optimization problems. J. Optim. Theory Appl. 119(2), 335–356 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Xu, Y.H., Liu, S.Y.: Benson proper efficiency in the nearly cone-subconvexlike vector optimization with set-valued functions. Appl. Math. J. Chin. Univ. Ser. B 18(1), 95–102 (2003)

    Article  MATH  Google Scholar 

  19. Borwein, J.M.: Properly efficient points for maximization with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)

    Article  MATH  Google Scholar 

  20. Shi, S.Z.: Convex Analysis. Universitext. Shanghai Scientifical and Technical Press, Shanghai (1990)

    Google Scholar 

  21. Tiel, J.V.: Convex Analysis. An Introductory Text. Wiley, New York (1984)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Z. A. Zhou.

Additional information

Communicated by J.-C. Yao.

The authors are grateful to Professor J.-C. Yao and the referees for valuable comments and suggestions.

X.M. Yang was supported by the National Nature Science Foundation of China (Grant 10831009).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, Z.A., Yang, X.M. Optimality Conditions of Generalized Subconvexlike Set-Valued Optimization Problems Based on the Quasi-Relative Interior. J Optim Theory Appl 150, 327–340 (2011). https://doi.org/10.1007/s10957-011-9844-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9844-0

Keywords

Navigation