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Efficiency and Duality in Nondifferentiable Multiobjective Programming Involving Directional Derivative

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DOI: 10.4236/am.2011.24057    6,880 Downloads   10,646 Views   Citations
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ABSTRACT

In this paper, we introduce a new class of generalized dI-univexity in which each component of the objective and constraint functions is directionally differentiable in its own direction di for a nondifferentiable multiobjective programming problem. Based upon these generalized functions, sufficient optimality conditions are established for a feasible point to be efficient and properly efficient under the generalised dI-univexity requirements. Moreover, weak, strong and strict converse duality theorems are also derived for Mond-Weir type dual programs.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

I. Ahmad, "Efficiency and Duality in Nondifferentiable Multiobjective Programming Involving Directional Derivative," Applied Mathematics, Vol. 2 No. 4, 2011, pp. 452-460. doi: 10.4236/am.2011.24057.

References

[1] M. A. Hanson, “On Sufficiency of the Kunn-Tucker Conditions,” Journal of Mathematical Analysis and Applications, Vol. 80, 1981, pp. 445-550.
[2] B. D. Craven, “Invex Functions and Constrained Local Minima,” Bulletin of Australian Mathematical Society, Vol. 24, No. 3, 1981, pp. 357-366. doi:10.1017/S0004972700004895
[3] R. N. Kaul and K. Kaur, “Optimality Criteria in Nonlinear Programming Involving Non Convex Functions,” Journal of Mathematical Analysis and Applications, Vol. 105, No. 1, January 1985, pp. 104-112. doi:10.1016/0022-247X(85)90099-X
[4] M. A. Hanson and B. Mond, “Necessary and Sufficient Conditions in Constrained Optimization,” Mathematical Programming, Vol. 37, No. 1, 1987, pp. 51-58. doi:10.1007/BF02591683
[5] N. G. Ruedo and M. A. Hanson, “Optimality Criteria in Mathematical Programming Involving Generalized Invexity,” Journal of Mathematical Analysis and Applications, Vol. 130, No. 2, 1988, pp. 375-385. doi:10.1016/0022-247X(88)90313-7
[6] F. Zhao, “On Sufficiency of the Kunn-Tucker Conditions in Non Differentiable Programming,” Bulletin Australian Mathematical Society, Vol. 46, No. 3, 1992, pp. 385-389.
[7] F. H. Clarke, “Optimization and Nonsmooth Analysis,” John Wiley and Sons, New York, 1983.
[8] R. N. Kaul, S. K. Suneja and M. K. Srivastava, “Optimality Criteria and Duality in Multi Objective Optimization Involving Generalized Invexity,” Journal of Optimization Theory and Applications, Vol. 80, No. 3, 1994, pp. 465-482. doi:10.1007/BF02207775
[9] S. K. Suneja and M. K. Srivastava, “Optimality and Duality in Non Differentiable Multi Objective Optimization Involving -Type I and Related Functions,” Journal of Mathematical Analysis and Applications, Vol. 206, 1997, pp. 465-479. doi:10.1006/jmaa.1997.5238
[10] H. Kuk and T. Tanino, “Optimality and Duality in Non-smooth Multi Objective Optimization Involving Generalized Type I Functions,” Computers and Mathematics with Applications, Vol. 45, No. 10-11, 2003, pp. 1497-1506. doi:10.1016/S0898-1221(03)00133-0
[11] T. R. Gulati and D. Agarwal, “Sufficiency and Duality in Nonsmooth Multiobjective Optimization Involving Generalized -Type I Functions,” Computers and Mathematics with Applications, Vol. 52, No. 1-2, July 2006, pp. 81-94. doi:10.1016/j.camwa.2006.08.006
[12] R. P. Agarwal, I. Ahmad, Z. Husain and A. Jayswal, “Optimality and Duality in Nonsmooth Multiobjective Optimization Involving Generalized -Type I Functions,” Journal of Inequalities and Applications, 2010. doi:10.1155/2010/898626
[13] A. Jayswal, I. Ahmad and S. Al-Homidan, “Sufficiency and Duality for Nonsmooth Multiobjective Programming Problems Involving Generalized -Univex Functions,” Mathematical and Computer Modelling, Vol. 53, 2011, pp. 81-90. doi:10.1016/j.mcm.2010.07.020
[14] T. Antczak, “Multiobjective Programming under -Invexity,” European Journal of Operational Research, Vol. 137, No. 1, 2002, pp. 28-36. doi:10.1016/S0377-2217(01)00092-3
[15] S. K. Mishra, S. Y. Wang and K. K. Lai, “Optimality and Duality in Nondifferentiable and Multi Objective Programming under Generalized -Invexity,” Journal of Global Optimization, Vol. 29, No. 4, 2004, pp. 425-438. doi:10.1023/B:JOGO.0000047912.69270.8c
[16] S. K. Mishra, S. Y. Wang and K. K. Lai, “Nondifferentiable Multiobjective Programming under Generalized -Univexity,” European Journal of Operational Research, Vol. 160, No. 1, 2005, pp. 218-226. doi:10.1016/S0377-2217(03)00439-9
[17] S. K Mishra and M. A. Noor, “Some Nondifferentiable Multiobjective Programming Problems,” Journal of Mathematical Analysis and Applications, Vol. 316, No. 2, April 2006, pp. 472-482. doi:10.1016/j.jmaa.2005.04.067
[18] Y. L. Ye, “ -Invexity and Optimality Conditions,” Journal of Mathematical Analysis and Applications, Vol. 162, No. 1, November 1991, pp. 242-249. doi:10.1016/0022-247X(91)90190-B
[19] T. Antczak, “Optimality Conditions and Duality for Nondifferentiable Multi Objective Programming Problems Involving -Type I Functions,” Journal of Computational and Applied Mathematics, Vol. 225, No. 1, March 2009, pp. 236-250. doi:10.1016/j.cam.2008.07.028
[20] H. Silmani and M. S. Radjef, “Nondifferentiable Multiobjective Programming under Generalized -Invexity,” European Journal of Operational Research, Vol. 202, 2010, pp. 32-41. doi:10.1016/j.ejor.2009.04.018
[21] R. P. Agarwal, I. Ahmad and S. Al-Homidan, “Optimality and Duality for Nonsmooth Multiobjective Programming Problems Involving Generalized -Type I Invex Functions,” Journal of Nonlinear and Convex Analysis, 2011.
[22] T. Antczak, “Mean Value in Invexity Analysis,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 60, No. 8, March 2005, pp. 1473-1484.
[23] A. Ben-Israel and B. Mond, “What is Invexity?” The Journal of Australian Mathematical Society Series B, Vol. 28, No. 1, 1986, pp. 1-9. doi:10.1017/S0334270000005142
[24] V. Jeyakumar and B. Mond, “On Generalized Convex Mathematical Programming,” Journal of the Australian Mathematical Society Series B, Vol. 34, 1992, pp. 43-53. doi:10.1017/S0334270000007372
[25] M. A. Hanson, R. Pini and C. Singh, “Multiobjective Programming under Generalized Type I Invexity,” Journal of Mathematical Analysis and Applications, Vol. 261, No. 2, September 2001, pp. 562-577. doi:10.1006/jmaa.2001.7542

  
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