Skip to main content
SpringerLink
Log in

Optimality conditions and duality on approximate solutions in vector optimization with arcwise connectivity

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper a new class of generalized vector-valued arcwise connected functions, termed sub-arcwise connected functions, is introduced. The properties of sub-arcwise connected functions are derived. The approximate quasi efficient solutions of vector optimization problems are studied, and the necessary and sufficient optimality conditions are obtained under the assumption of arcwise connectivity. An approximate Mond-Weir type dual problem is formulated and the duality theorems are established.

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

Access this article

Log in via an institution

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. Avriel M., Zang I.: Generalized arcwise connected functions and characterization of local-global minimum properties. J. Optim. Theory Appl. 32, 407–425 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bhatia D., Mehra A.: Optimality conditions and duality inuvolving arcwise connected and generalized arcwise connected functions. J. Optim. Theory Appl. 100, 181–194 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carven B.D.: Mathematics Programming and Control Theory. Chapman and Hall, London (1978)

    Book  Google Scholar 

  4. Dutta J.: Necessary optimality conditions and saddle points for approximate optimization in Banach spaces. Top 13, 127–143 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dutta J., Vetrivel V.: On approximate minima in vector optimization. Numer. Funct. Anal. Optim. 22, 845–859 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fu J.Y., Wang H.: Arcwise connected cone-convex functions and mathematical programming. J. Optim. Theory Appl. 118, 339–352 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gutiérrez C., Loépez R., Novo V.: Generalized ε-quasi-solutions in multiobjective optimization problems: existence results and optimality conditions. Nonlinear Anal. 72, 4331–4346 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Helbig, S.: On a New Concept for ε-Efficiency. talk at “Optimization Days 1992”, Montreal (1992)

  9. Jahn J.: Mathematical Vector Optimization in Partially-Ordered Linear Spaces. Peter Lang, Frankfurt am Main (1986)

    MATH  Google Scholar 

  10. Jeyakumar V.: Convexlike alternative theorems and mathematical programming. Optimization 16, 643–652 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kutateladze S.S.: Convex ε-programming. Soviet Math. Dokl. 20, 391–393 (1979)

    MATH  Google Scholar 

  12. Liu, C.P., Yang, X.M., Lee, H.W.J.: Lagrange multiplier rules for approximate solutions in vector optimization. submitted to J. Math. Anal. Appl.

  13. Loridan P.: Necessary conditions for ε-optimality. Math. Program. Stud. 19, 140–152 (1982)

    Article  MathSciNet  Google Scholar 

  14. Loridan P.: ε-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mukherjee R.N., Yadav S.R.: A note on arcwise connected sets and functions. B. Aust. Math. Soc. 31, 369–375 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Németh A.B.: A nonconvex vector minimization problem. Nonlinear Anal. 10, 669–678 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ortega J.M., Rheinboldt W.C.: Iterative Solutions of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    Google Scholar 

  18. Singh G.: Elementary properties of arcwise connected sets and functions. J. Optim. Theory Appl. 41, 377–387 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Suneja S.K., Aggarwal S., Davar S.: Generalized connected functions with respect to cones. J. Optim Theory Appl. 106, 399–410 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tanaka T.: A new approach to approximation of solutions in vector optimization problems. In: Fushimi, M., Tone, K. (eds) Proceedings of APORS, pp. 497–504. World Scientific Publishing, Singapore (1995)

    Google Scholar 

  21. White D.J.: Epsilon efficiency. J. Optim. Theory Appl. 49, 319–337 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang Q.X.: Optimality conditions and duality for semi-infinite programming involving B-arcwise connected functions. J. Glob. Optim. 45, 615–629 (2009)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    C. P. Liu

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

    H. W. J. Lee

  3. Department of Mathematics, Chongqing Normal University, Chongqing, 400047, China

    X. M. Yang

Authors
  1. C. P. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. W. J. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. X. M. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X. M. Yang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, C.P., Lee, H.W.J. & Yang, X.M. Optimality conditions and duality on approximate solutions in vector optimization with arcwise connectivity. Optim Lett 6, 1613–1626 (2012). https://doi.org/10.1007/s11590-011-0352-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-011-0352-9

Keywords

Search

Navigation