On Second-Order Duality in Nondifferentiable Continuous Programming

Affiliation(s)

Department of Mathematics, Jaypee University of Engineering and Technology, Guna (M.P), India.

Department of Mathematics, Jaypee University of Engineering and Technology, Guna (M.P), India.

ABSTRACT

A Mond-Weir type second-order dual continuous programming problem associated with a class of nondifferentiable continuous programming problems is formulated. Under second-order pseudo-invexity and second-order quasi-invexity various duality theorems are established for this pair of dual continuous programming problems. A pair of dual continuous programming problems with natural boundary values is constructed and the proofs of its various duality results are briefly outlined. Further, it is shown that our results can be regarded as dynamic generalizations of corresponding (static) second-order duality theorems for a class of nondifferentiable nonlinear programming problems already studied in the literature.

A Mond-Weir type second-order dual continuous programming problem associated with a class of nondifferentiable continuous programming problems is formulated. Under second-order pseudo-invexity and second-order quasi-invexity various duality theorems are established for this pair of dual continuous programming problems. A pair of dual continuous programming problems with natural boundary values is constructed and the proofs of its various duality results are briefly outlined. Further, it is shown that our results can be regarded as dynamic generalizations of corresponding (static) second-order duality theorems for a class of nondifferentiable nonlinear programming problems already studied in the literature.

Cite this paper

I. Husain and S. Shrivastav, "On Second-Order Duality in Nondifferentiable Continuous Programming,"*American Journal of Operations Research*, Vol. 2 No. 3, 2012, pp. 289-295. doi: 10.4236/ajor.2012.23035.

I. Husain and S. Shrivastav, "On Second-Order Duality in Nondifferentiable Continuous Programming,"

References

[1] X. Chen, “Second-Order Duality for Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 286, 2003, pp. 261-270. doi:10.1016/S0022-247X(03)00481-5

[2] I. Husain, A. Ahmed and M. Masoodi, “Second-Order Duality for Variational Problems,” European Journal of Pure and Applied Mathematics, Vol. 2, No. 2, 2009, pp. 278-295.

[3] B. Mond and T. Weir, “Generalized Convexity and High Order-Duality,” Journal of Mathematical Analysis and Applications, Vol. 46, No. 1, 1974, pp. 169-174. doi:10.1016/0022-247X(74)90289-3

[4] I. Husain and M. Masoodi, “Second-Order Duality for a Class of Nondifferentiable Continuous Programming Problems,” European Journal of Pure and Applied Mathematics, Vol. 5, No. 3, 2012, pp. 390-400.

[5] J. Zhang and B. Mond, “Duality for a Nondifferentiable Programming Problem,” Bulletin Australian Mathematical Society, Vol. 55, 1997, pp. 20-44.

[6] S. Chandra, B. D. Craven and I. Husain, “A Class of Nondifferentiable continuous Programming Problems,” Journal of Mathematical Analysis and Applications, Vol. 107, No. 1, 1985, pp. 122-131. doi:10.1016/0022-247X(85)90357-9

[1] X. Chen, “Second-Order Duality for Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 286, 2003, pp. 261-270. doi:10.1016/S0022-247X(03)00481-5

[2] I. Husain, A. Ahmed and M. Masoodi, “Second-Order Duality for Variational Problems,” European Journal of Pure and Applied Mathematics, Vol. 2, No. 2, 2009, pp. 278-295.

[3] B. Mond and T. Weir, “Generalized Convexity and High Order-Duality,” Journal of Mathematical Analysis and Applications, Vol. 46, No. 1, 1974, pp. 169-174. doi:10.1016/0022-247X(74)90289-3

[4] I. Husain and M. Masoodi, “Second-Order Duality for a Class of Nondifferentiable Continuous Programming Problems,” European Journal of Pure and Applied Mathematics, Vol. 5, No. 3, 2012, pp. 390-400.

[5] J. Zhang and B. Mond, “Duality for a Nondifferentiable Programming Problem,” Bulletin Australian Mathematical Society, Vol. 55, 1997, pp. 20-44.

[6] S. Chandra, B. D. Craven and I. Husain, “A Class of Nondifferentiable continuous Programming Problems,” Journal of Mathematical Analysis and Applications, Vol. 107, No. 1, 1985, pp. 122-131. doi:10.1016/0022-247X(85)90357-9