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Optimality and duality in vector optimization involving generalized type I functions over cones

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Abstract

In this paper generalized type-I, generalized quasi type-I, generalized pseudo type-I and other related functions over cones are defined for a vector minimization problem. Sufficient optimality conditions are studied for this problem using Clarke’s generalized gradients. A Mond-Weir type dual is formulated and weak and strong duality results are established.

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References

  1. Antzak T.: Saddle point criteria in an η-approximation method for nonlinear mathematical programming problems involving invex functions. J. Optim. Theory Appl. 132, 71–87 (2007). doi:10.1007/s10957-006-9069-9

    Article  Google Scholar 

  2. Bector C.R., Chandra S., Bector M.K.: Sufficient optimality conditions and duality for a quasiconvex programming problem. J. Optim. Theory Appl. 59, 209–221 (1988)

    Google Scholar 

  3. Clarke F.H.: Optimization and nonsmooth analysis. Wiley, New York (1983)

    Google Scholar 

  4. Craven B.D.: Non differentiable optimization by smooth approximations. Optimization 17, 3–17 (1986). doi:10.1080/02331938608843097

    Article  Google Scholar 

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

    Article  Google Scholar 

  6. Hanson M.A.: On sufficiency of the Kuhn Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981). doi:10.1016/0022-247X(81)90123-2

    Article  Google Scholar 

  7. Hanson, M.A., Mond, B.: Necessary and sufficient conditions in constrained optimization, FSU statistics report, H683, Florida State University, Department of Statistics, Tallhassae, Florida (1984)

  8. Kaul R.N., Kaur S.: Optimality criteria in nonlinear programming involving nonconvex functions. J. Math. Anal. Appl. 105, 104–112 (1985). doi:10.1016/0022-247X(85)90099-X

    Article  Google Scholar 

  9. Khurana S.: Symmetric duality in multiobjective programming involving generalized cone-invex functions. Eur. J. Oper. Res. 165, 592–597 (2005). doi:10.1016/j.ejor.2003.03.004

    Article  Google Scholar 

  10. Kuk H., Tanino T.: Optimality and duality in nonsmooth multiobjecive optimization involving generalized type-I functions. Comput. Math. Appl. 45, 1497–1506 (2003). doi:10.1016/S0898-1221(03)00133-0

    Article  Google Scholar 

  11. Mangasarain O.L.: Non linear programming. McGraw Hill, New York (1969)

    Google Scholar 

  12. Nobakhtian S.: Sufficiency in nonsmooth multiobjective programming involving generalized (F, ρ)-convexity. J. Optim. Theory Appl. 130, 359–365 (2006). doi:10.1007/s10957-006-9105-9

    Article  Google Scholar 

  13. Reiland T.W.: Generalized invexity for nonsmooth vector valued mappings. Numer. Funct. Anal. Optim. 10, 1191–1202 (1989). doi:10.1080/01630568908816352

    Article  Google Scholar 

  14. Rueda N.G., Hanson M.A.: Optimality criteria in mathematical programming involving generalized invexity. J. Math. Anal. Appl. 130, 375–385 (1988). doi:10.1016/0022-247X(88)90313-7

    Article  Google Scholar 

  15. Sach P.H., Kim D.S., Lee G.M.: Invexity as necessary optimality condition in nonsmooth programs. J. Korean Math. Soc. 43, 241–258 (2006)

    Article  Google Scholar 

  16. Soleimani-damaneh M.: Characterization of nonsmooth quasiconvex and pseudoconvex function. J. Math. Anal. Appl. 330, 1387–1392 (2007). doi:10.1016/j.jmaa.2006.08.033

    Article  Google Scholar 

  17. Suneja S.K., Khurana S., Vani: Generalized nonsmooth invexity over cones in vector optimization. Eur. J. Oper. Res. 186, 28–40 (2008). doi:10.1016/j.ejor.2007.01.047

    Article  Google Scholar 

  18. Suneja S.K., Lalitha C.S., Khurana S.: Optimality and duality theorems for nonsmooth multiobjective fractional programs. Indian J. Pure App. Math. 30, 243–257 (1999)

    Google Scholar 

  19. Yen N.D., Sach P.H.: On locally lipschitz vector valued invex functions. Bull. Aust. Math. Soc. 47, 259–271 (1993)

    Article  Google Scholar 

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

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Miranda House, University of Delhi, Delhi, 110007, India

    S. K. Suneja

  2. Department of Mathematics, Daulat Ram College, University of Delhi, Delhi, 110007, India

    Seema Khurana

  3. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Meetu Bhatia

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  1. S. K. Suneja
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  2. Seema Khurana
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  3. Meetu Bhatia
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Correspondence to Meetu Bhatia.

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Suneja, S.K., Khurana, S. & Bhatia, M. Optimality and duality in vector optimization involving generalized type I functions over cones. J Glob Optim 49, 23–35 (2011). https://doi.org/10.1007/s10898-009-9522-z

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  • DOI: https://doi.org/10.1007/s10898-009-9522-z

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