On Continuous Programming with Support Functions

Affiliation(s)

Department of Mathematics, Jaypee University of Engineering and Technology, Guna, India.

Department of Statistics, University of Kashmir, Srinagar, India.

Department of Mathematics, Jaypee University of Engineering and Technology, Guna, India.

Department of Statistics, University of Kashmir, Srinagar, India.

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.

Husain, I. , Shrivastav, S. and Shah, A. (2013) On Continuous Programming with Support Functions.

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

[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