Skip to main content
SpringerLink
Log in

Nonsmooth multiobjective optimization involving generalized univex functions

  • Theoretical Article
  • Published:
OPSEARCH Aims and scope Submit manuscript

Abstract

In this paper, we have considered a class of constrained nonsmooth multi objective programming problem involving semi-directionally differentiable functions from a view point of generalized convexity. A new generalized class of (d I —ρ—σ)-V-type I univex functions is introduced under which various weak, strong, converse and strict converse duality theorems are established for Mond-Weir type dual program in order to relate the efficient and weak efficient solutions of primal and dual problem. Also, we have illustrated through various non-trivial examples that this class extends many earlier studied classes in literature.

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. Mangasarian, O.L.: Nonlinear Programming. AZ McGrawHill, New York (1969)

    Google Scholar 

  2. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd ed.Wiley, New York (2006)

    Book  Google Scholar 

  3. Hanson, M.A.: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)

    Article  Google Scholar 

  4. Craven, B.D.: Invex functions and constrained local minima. Bull. Aust. Math. Soc. 24, 357–366 (1981)

    Article  Google Scholar 

  5. Hanson, M.A., Pini, R., Singh, C.: Multiobjective programming under generalized type I invexity. J. Math. Anal. Appl. 261, 562–577 (2001)

    Article  Google Scholar 

  6. Zhao, F.: On sufficiency of the Kuhn-Tucker conditions in nondifferentiable programming. Bull. Aust. Math. Soc. 46, 385–389 (1992)

    Article  Google Scholar 

  7. Antczak, T.: Multiobjective programming under d-invexity. Eur. J. Oper. Res. 137, 28–36 (2002)

    Article  Google Scholar 

  8. Ye, Y.L.: d-invexity and optimality conditions. J. Math. Anal. Appl. 162, 242–249 (1991)

    Article  Google Scholar 

  9. Nahak, C., Mohapatra, R.N.: \(d-\rho -\eta -\theta \) invexity in multiobjective optimization. Nonlinear Anal. 70, 2288–2296 (2009)

    Article  Google Scholar 

  10. Suneja, S.K., Srivastava, M.K.: Optimality and duality in nondifferentiable multiobjective optimization involving d-type I and related functions. J. Math. Anal. Appl. 206, 465–479 (1997)

    Article  Google Scholar 

  11. Slimani, H., Radjef, M.S.: Nondifferentiable multiobjective programming under generalized \(d_{I}\) invexity. Eur. J. Oper. Res. 202, 32–41 (2010)

    Article  Google Scholar 

  12. Bector, C.R., Suneja, S.K., Gupta, S.: Univex functions and univex nonlinear programming. In: Proceeding of the administrative sciences association of Canada, pp. 115–124 (1992)

  13. Rueda, N.G., Hanson, M.A., Singh C.: Optimality and duality with generalized convexity. J. Optim. Theory Appl. 86, 491–500 (1995)

    Article  Google Scholar 

  14. Mishra, S.K.: On multiple-objective optimization with generalized univexity. J. Math. Anal. Appl. 224, 131–148 (1998)

    Article  Google Scholar 

  15. Mishra, S.K., Wang, S.Y., Lai, K.K.: Nondifferentiable multiobjective programming under generalized d-univexity. Eur. J. Oper. Res. 160, 218–226 (2005)

    Article  Google Scholar 

  16. Ahmad, I.: Efficiency and duality in nondifferentiable multiobjective programs involving directional derivative. Appl. Math. 2, 452–460 (2011)

    Article  Google Scholar 

  17. Hanson, M.A., Mond, B.: Necessary and sufficient conditions in constrained optimization. Math. Program. 37, 51–58 (1987)

    Article  Google Scholar 

  18. Mishra, S.K., Noor, M.A.: Some nondifferentiable multiobjective programming problems. J. Math. Anal. Appl. 316, 472–482 (2006)

    Article  Google Scholar 

  19. Mishra, S.K., Wang, S.Y., Lai, K.K.: Optimality and duality in nondifferentiable and multiobjective programming under generalized d-invexity. J Glob. Optim. 29, 425–438 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Mathematical Sciences, Banasthali University, Rajasthan, India

    Pallavi Kharbanda

  2. Department of Applied Sciences and Humanities, ITM University, Gurgaon, India

    Divya Agarwal

  3. Department of Mathematics, South Asian University, New Delhi, India

    Deepa Sinha

Authors
  1. Pallavi Kharbanda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Divya Agarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Deepa Sinha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pallavi Kharbanda.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kharbanda, P., Agarwal, D. & Sinha, D. Nonsmooth multiobjective optimization involving generalized univex functions. OPSEARCH 51, 130–147 (2014). https://doi.org/10.1007/s12597-013-0135-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12597-013-0135-4

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation