Multiobjective Nonlinear Symmetric Duality Involving Generalized Pseudoconvexity

Author(s)
Mohamed Abd El-Hady Kassem

ABSTRACT

The purpose of this paper is to introduce second order (*K*, *F*)-pseudoconvex and second order strongly (*K*, *F*)- pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex functions. A pair of second order symmetric dual multiobjective nonlinear programs is formulated by using the considered functions. Furthermore, the weak, strong and converse duality theorems for this pair are established. Finally, a self duality theorem is given.

The purpose of this paper is to introduce second order (

KEYWORDS

Multiobjective Programming, Second-Order Symmetric Dual Models, Duality Theorems, Pseudoconvex Functions, Cones

Multiobjective Programming, Second-Order Symmetric Dual Models, Duality Theorems, Pseudoconvex Functions, Cones

Cite this paper

nullM. Kassem, "Multiobjective Nonlinear Symmetric Duality Involving Generalized Pseudoconvexity,"*Applied Mathematics*, Vol. 2 No. 10, 2011, pp. 1236-1242. doi: 10.4236/am.2011.210172.

nullM. Kassem, "Multiobjective Nonlinear Symmetric Duality Involving Generalized Pseudoconvexity,"

References

[1] W. S. Dorn, “A Symmetric Dual Theorem for Quadratic Programs,” Journal of the Operations Research Society of Japan, Vol. 2, 1960, pp. 93-97.

[2] G. B. Dantzig, E. Eisenberg and R. W. Cottle, “Symmetric Dual Nonlinear Programs,” Pacific Journal of Mathematics, Vol. 15, No. 3, 1965, pp. 809-812.

[3] M. S. Bazaraa and J. J. Goode, “On Symmetric Duality in Nonlinear Programming,” Operation Research, Vol. 21, No. 1, 1973, pp. 1-9. doi:10.1287/opre.21.1.1

[4] D. Sang Kim, Y. B. Yun and W. J. Lee, “Multiobjective Symmetric Duality with Cone Constraints,” European Journal of Operational Research, Vol. 107, No. 3, 1998, pp. 686-691. doi:10.1016/S0377-2217(97)00322-6

[5] S. H. Hou and X. M. Yang, “On Second-Order Symmetric Duality in Non-Differentiable Programming,” Journal of Mathe-matical Analysis and Applications, Vol. 255, 2001, pp. 491-498. doi:10.1006/jmaa.2000.7242

[6] X. M. Yang, X. Q. Yang and K. L. Teo, “Non-Differentiable Second Order Symmetric Duality in Mathematical Programming with F-Convexity,” European Journal of Operational Research, Vol. 144, 2003, pp. 554-559. doi:10.1016/S0377-2217(02)00156-X

[7] X. M. Yang, X. Q. Yang and K. L. Teo, “Converse Duality in Nonli-near Programming with Cone Constraints,” European Journal of Operational Research, Vol. 170, 2006, pp. 350-354. doi:10.1016/j.ejor.2004.05.028

[8] X. M. Yang, X. Q. Yang, K. L. Teo and S. H. Hou, “Multiobjective Second Order Symmetric with F-Convexity,” European Journal of Operational Research, Vol. 165, No. 3, 2005, pp. 585-591. doi:10.1016/j.ejor.2004.01.028

[9] K. Suneja, S. Aggarwal and S. Davar, “Multiobjective Sym-metric Duality involving Cones,” European Journal of Operational Research, Vol. 141, No. 3, 2002, pp. 471-479. doi:10.1016/S0377-2217(01)00258-2

[10] S. Khurana, “Symmetric Duality in Multiobjective Programming involving Generalized Cone-Invex Functions,” European Journal of Operational Research, Vol. 165, No. 3, 2005, pp. 592-597. doi:10.1016/j.ejor.2003.03.004

[11] S. Chandra and A. Abha, “A Note on Pseudo-Invex and Duality in Nonlinear Programming,” European Journal of Operational Research, Vol. 122, No. 1, 2000, pp. 161-165. doi:10.1016/S0377-2217(99)00076-4

[12] M. Kassem, “Higher-Order Symmetric Duality in Vector Optimization Problem involving Generalized Cone-Invex Functions,” Applied Mathematics and Computation, Vol. 209, No. 2, 2009, pp. 405-409. doi:10.1016/j.amc.2008.12.063

[1] W. S. Dorn, “A Symmetric Dual Theorem for Quadratic Programs,” Journal of the Operations Research Society of Japan, Vol. 2, 1960, pp. 93-97.

[2] G. B. Dantzig, E. Eisenberg and R. W. Cottle, “Symmetric Dual Nonlinear Programs,” Pacific Journal of Mathematics, Vol. 15, No. 3, 1965, pp. 809-812.

[3] M. S. Bazaraa and J. J. Goode, “On Symmetric Duality in Nonlinear Programming,” Operation Research, Vol. 21, No. 1, 1973, pp. 1-9. doi:10.1287/opre.21.1.1

[4] D. Sang Kim, Y. B. Yun and W. J. Lee, “Multiobjective Symmetric Duality with Cone Constraints,” European Journal of Operational Research, Vol. 107, No. 3, 1998, pp. 686-691. doi:10.1016/S0377-2217(97)00322-6

[5] S. H. Hou and X. M. Yang, “On Second-Order Symmetric Duality in Non-Differentiable Programming,” Journal of Mathe-matical Analysis and Applications, Vol. 255, 2001, pp. 491-498. doi:10.1006/jmaa.2000.7242

[6] X. M. Yang, X. Q. Yang and K. L. Teo, “Non-Differentiable Second Order Symmetric Duality in Mathematical Programming with F-Convexity,” European Journal of Operational Research, Vol. 144, 2003, pp. 554-559. doi:10.1016/S0377-2217(02)00156-X

[7] X. M. Yang, X. Q. Yang and K. L. Teo, “Converse Duality in Nonli-near Programming with Cone Constraints,” European Journal of Operational Research, Vol. 170, 2006, pp. 350-354. doi:10.1016/j.ejor.2004.05.028

[8] X. M. Yang, X. Q. Yang, K. L. Teo and S. H. Hou, “Multiobjective Second Order Symmetric with F-Convexity,” European Journal of Operational Research, Vol. 165, No. 3, 2005, pp. 585-591. doi:10.1016/j.ejor.2004.01.028

[9] K. Suneja, S. Aggarwal and S. Davar, “Multiobjective Sym-metric Duality involving Cones,” European Journal of Operational Research, Vol. 141, No. 3, 2002, pp. 471-479. doi:10.1016/S0377-2217(01)00258-2

[10] S. Khurana, “Symmetric Duality in Multiobjective Programming involving Generalized Cone-Invex Functions,” European Journal of Operational Research, Vol. 165, No. 3, 2005, pp. 592-597. doi:10.1016/j.ejor.2003.03.004

[11] S. Chandra and A. Abha, “A Note on Pseudo-Invex and Duality in Nonlinear Programming,” European Journal of Operational Research, Vol. 122, No. 1, 2000, pp. 161-165. doi:10.1016/S0377-2217(99)00076-4

[12] M. Kassem, “Higher-Order Symmetric Duality in Vector Optimization Problem involving Generalized Cone-Invex Functions,” Applied Mathematics and Computation, Vol. 209, No. 2, 2009, pp. 405-409. doi:10.1016/j.amc.2008.12.063