Efficiency and Duality in Nondifferentiable Multiobjective Programming Involving Directional Derivative

Author(s)
Izhar Ahmad

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.

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.

KEYWORDS

Multiobjective Programming, Nondifferentiable Programming, Generalized dI-Univexity, Sufficiency, Duality

Multiobjective Programming, Nondifferentiable Programming, Generalized dI-Univexity, Sufficiency, Duality

Cite this paper

nullI. 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.

nullI. Ahmad, "Efficiency and Duality in Nondifferentiable Multiobjective Programming Involving Directional Derivative,"

References

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[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

[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