Skip to main content
Log in

Optimality conditions and duality for interval-valued optimization problems using convexifactors

  • Published:
Rendiconti del Circolo Matematico di Palermo (1952 -) Aims and scope Submit manuscript

Abstract

This paper is devoted to the applications of convexifactors on interval-valued programming problem. Based on the concept of LU optimal solution, sufficient optimality conditions are established under generalized \(\partial ^{*}\)-convexity assumptions. Furthermore, appropriate duality theorems are derived for two types of dual problem, namely Mond–Weir and Wolfe type duals. We also construct examples to manifest the established relations.

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. Ahmad, I., Jayswal, A., Banerjee, J.: On interval-valued optimization problems with generalized invex functions. J. Inequal. Appl. 2013, 313 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ansari Ardali, A., Movahedian, N., Nobakhtian, S.: Optimality conditions for nonsmooth mathematical programs with equilibrium constraints, using convexifactors. Optimization (2014). doi:10.1080/02331934.2014.987776

  3. Babahadda, H., Gadhi, N.: Necessary optimality conditions for bilevel optimization problems using convexificators. J. Glob. Optim. 34(4), 535–549 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bhurjee, A.K., Panda, G.: Sufficient optimality conditions and duality theory for interval optimization problem. Ann. Oper. Res. (2014). doi:10.1007/s10479-014-1644-0

  5. Bhurjee, A.K., Panda, G.: Multi-objective interval fractional programming problems: an approach for obtaining efficient solutions. Opsearch 52(1), 156–167 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  7. Demyanov, V.F.: Convexification and concavification of a positively homogeneous function by the same family of linear functions, Report 3, 208, 802, Universita di Pisa (1994)

  8. Demyanov, V.F., Jeyakumar, V.: Hunting for a smaller convex subdifferential. J. Glob. Optim. 10(3), 305–326 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dutta, J., Chandra, S.: Convexifactors, generalized convexity and optimality conditions. J. Optim. Theory Appl. 113(1), 41–64 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dutta, J., Chandra, S.: Convexifactors, generalized convexity and vector optimization. Optimization 53(1), 77–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gadhi, N.: Necessary and sufficient optimality conditions for fractional multi-objective problems. Optimization 57(4), 527–537 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Golestani, M., Nobakhtian, S.: Convexifactors and strong Kuhn–Tucker conditions. Comput. Math. Appl. 64(4), 550–557 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Golestani, M., Nobakhtian, S.: Optimality conditions for nonsmooth semidefinite programming via convexificators. Positivity 19(2), 221–336 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jayswal, A., Stancu-Minasian, I.M., Ahmad, I.: On sufficiency and duality for a class of interval-valued programming problems. Appl. Math. Comput. 218(8), 4119–4127 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jeyakumar, V., Luc, D.T.: Nonsmooth calculus, minimality, and monotonicity of convexificators. J. Optim. Theory Appl. 101(3), 599–621 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kohli, B.: Optimality conditions for optimistic bilevel programming problem using convexifactors. J. Optim. Theory Appl. 152(3), 632–651 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, X.F., Zhang, J.Z.: Necessary optimality conditions in terms of convexificators in Lipschitz optimization. J. Optim. Theory Appl. 131(3), 429–452 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Long, H.-J., Huang, N.-J.: Optimality conditions for efficiency on nonsmooth multiobjective programming problems. Taiwanese J. Math. 18(3), 687–699 (2014)

    Article  MathSciNet  Google Scholar 

  19. Luu, D.V.: Convexificators and necessary conditions for efficiency. Optimization 63(3), 321–335 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Luu, D.V.: Necessary and sufficient conditions for efficiency via convexificators. J. Optim. Theory Appl. 160(2), 510–526 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Michel, P., Penot, J.-P.: A generalized derivative for calm and stable functions. Diff. Integral Equ. 5(2), 433–454 (1992)

    MathSciNet  MATH  Google Scholar 

  22. Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)

    Book  MATH  Google Scholar 

  23. Mordukhovich, B.S., Shao, Y.: On nonconvex subdifferential calculus in Banach spaces. J. Convex Anal. 2(1), 211–228 (1995)

    MathSciNet  MATH  Google Scholar 

  24. Suneja, S.K., Kohli, B.: Duality for multiobjective fractional programming problem using convexifactors. Math. Sci. 7, 1–8, Art. No.: 6 (2013)

  25. Suneja, S.K., Kohli, B.: Generalized nonsmooth cone convexity in terms of convexifactors in vector optimization. Opsearch 50(1), 89–105 (2013)

    Article  MathSciNet  Google Scholar 

  26. Treiman, J.S.: The linear nonconvex generalized gradient and Lagrange multipliers. SIAM J. Optim. 5(3), 670–680 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, X., Jeyakumar, V.: A sharp Lagrangian multiplier rule for nonsmooth mathematical programming problems involving equality constraints. SIAM J. Optim. 10(4), 1136–1149 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wu, H.-C.: On interval-valued nonlinear programming problems. J. Math. Anal. Appl. 338(1), 299–316 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhou, H.-C., Wang, Y.-J.: Optimality condition and mixed duality for interval-valued optimization. In: Cao, B., Li, T.-F., Zhang, C.-Y. (eds.) Fuzzy Information and Engineering, Vol. 2. Advances in Intelligent and Soft Computing, vol. 62. Proceedings of the Third International Conference on Fuzzy Information and Engineering (ICFIE 2009), pp. 1315–1323. Springer, Berlin (2009)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anurag Jayswal.

Additional information

The research of the first author is financially supported by the DST, New Delhi, India through Grant no.: SR/FTP/MS-007/2011.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jayswal, A., Stancu-Minasian, I. & Banerjee, J. Optimality conditions and duality for interval-valued optimization problems using convexifactors. Rend. Circ. Mat. Palermo 65, 17–32 (2016). https://doi.org/10.1007/s12215-015-0215-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-015-0215-9

Keywords

Mathematics Subject Classification

Navigation