AM  Vol.4 No.10 , October 2013
On Continuous Programming with Support Functions
ABSTRACT

A second-order Mond-Weir type dual problem is formulated for a class of continuous programming problems in which both objective and constraint functions contain support functions; hence it is nondifferentiable. Under second-order strict pseudoinvexity, second-order pseudoinvexity and second-order quasi-invexity assumptions on functionals, weak, strong, strict converse and converse duality theorems are established for this pair of dual continuous programming problems. Special cases are deduced and a pair of dual continuous problems with natural boundary values is constructed. A close relationship between the duality results of our problems and those of the corresponding (static) nonlinear programming problem with support functions is briefly outlined.


Cite this paper
Husain, I. , Shrivastav, S. and Shah, A. (2013) On Continuous Programming with Support Functions. Applied Mathematics, 4, 1441-1449. doi: 10.4236/am.2013.410194.
References
[1]   X. H. Chen, “Second Order Duality for the Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 286, No. 1, 2003, pp. 261-270. http://dx.doi.org/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]   I. Husain and M. Masoodi, “Second-Order Duality for a Class of Nondifferentiable Continuous Programming Problems,” European Journal of Pure & Applied Mathematics, Vol. 5, No. 3, 2012, pp. 390-400.

[4]   I. Husain and S. K. Shrivastav, “On Second-Order Duality in Nondifferentiable Continuous Programming,” American Journal of Operations Research, Vol. 2, No. 3, 2012, pp. 289-295.
http://dx.doi.org/10.4236/ajor.2012.23035

[5]   O. L. Mangasarian, “Second and Higher Order Duality in Nonlinear Programming,” Journal of Mathematical Analysis and Applications, Vol. 51, No. 3, 1979, pp. 605-620.
http://dx.doi.org/10.1016/0022-247X(75)90111-0

[6]   I. Husain and M. Masoodi “Second-Order Duality for Continuous Programming Containing support Functions,” Applied Mathematics, Vol. 1, No. 6, 2010, pp. 534-541.
http://dx.doi.org/10.4236/am.2010.16071

[7]   I. Husain, A. Ahmed and M. Masoodi, “Second Order Duality in Mathematical Programming with Support Functions,” Journal of Informatics and Mathematical Sciences, Vol. 1, No. 2-3, 2009, pp. 165-182.

[8]   I. Husain and Z. Jabeen, “Continuous Programming Containing Support Functions,” Journal of Applied Mathematics & Informatics, Vol. 26, No. 1-2, 2008, pp. 75-106.

[9]   B. Mond and M. Schechter, “Nondifferentiable Symmetric Duality,” Bulletin of the Australian Mathematical Society, Vol. 53, 1996, pp. 177-188. http://dx.doi.org/10.1017/S0004972700016890

[10]   M. Schechter, “More on Subgradient Duality,” Journal of Mathematical Analysis and Applications, Vol. 71, No. 1, 1979, pp. 251-262. http://dx.doi.org/10.1016/0022-247X(79)90228-2

 
 
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