Second-Order Duality for Continuous Programming Containing Support Functions

ABSTRACT

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

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

KEYWORDS

Continuous Programming, Second-Order Invexity, Second-Order Pseudoinvexity, Second-Order Duality, Nonlinear Programming, Support Functions, Natural Boundary Values

Continuous Programming, Second-Order Invexity, Second-Order Pseudoinvexity, Second-Order Duality, Nonlinear Programming, Support Functions, Natural Boundary Values

Cite this paper

nullI. Husain and M. Masoodi, "Second-Order Duality for Continuous Programming Containing Support Functions,"*Applied Mathematics*, Vol. 1 No. 6, 2010, pp. 534-541. doi: 10.4236/am.2010.16071.

nullI. Husain and M. Masoodi, "Second-Order Duality for Continuous Programming Containing Support Functions,"

References

[1] O. L. Mangasarian, “Second and Higher Order Duality in Non linear Programming,” Journal of Mathematical Analysis and Applications, Vol. 51, 1979, pp. 605-620.

[2] B. Mond,“Second Order Duality in Non-Linear Programming,” Opsearch, Vol. 11, 1974, pp. 90-99.

[3] B. Mond and T. Weir, “Generalized Convexity and Higher Order Duality,” Journal of Mathematical Analysis and Applications, Vol. 46, 1974, pp. 169-174.

[4] B. Mond and M. A. Hanson, “Duality for Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 11, 1965, pp. 355-364.

[5] C. R. Bector, S. Chandra and I. Husain, “Generalized Concavity and Duality in Continuous Programming,” Utilitas Mathematica, Vol. 25, 1984.

[6] B. Mond and I. Husain, “Sufficient Optimality Criteria and Duality for Avariational Problem with Generalized Invexity,” The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, Vol. 31, No. 1, 1989, pp. 101-121.

[7] X. Chen, “Second-Order Duality for the Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 216, 2003, pp. 261-270.

[8] 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. 271- 295.

[9] I. Husain and M. Masoodi, “Second-Order Duality for a Class of Nondifferentiable Continuous Programming Problems,” European Journal of Pure and Applied Mathematics, 2010 (in press).

[10] 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.

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

[12] B. Mond and M. Schechter, “Nondifferentiable Symmetric Duality,” Bulletin of the Australian Mathematical Society, Vol. 53, 1996, pp. 177-188.

[1] O. L. Mangasarian, “Second and Higher Order Duality in Non linear Programming,” Journal of Mathematical Analysis and Applications, Vol. 51, 1979, pp. 605-620.

[2] B. Mond,“Second Order Duality in Non-Linear Programming,” Opsearch, Vol. 11, 1974, pp. 90-99.

[3] B. Mond and T. Weir, “Generalized Convexity and Higher Order Duality,” Journal of Mathematical Analysis and Applications, Vol. 46, 1974, pp. 169-174.

[4] B. Mond and M. A. Hanson, “Duality for Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 11, 1965, pp. 355-364.

[5] C. R. Bector, S. Chandra and I. Husain, “Generalized Concavity and Duality in Continuous Programming,” Utilitas Mathematica, Vol. 25, 1984.

[6] B. Mond and I. Husain, “Sufficient Optimality Criteria and Duality for Avariational Problem with Generalized Invexity,” The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, Vol. 31, No. 1, 1989, pp. 101-121.

[7] X. Chen, “Second-Order Duality for the Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 216, 2003, pp. 261-270.

[8] 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. 271- 295.

[9] I. Husain and M. Masoodi, “Second-Order Duality for a Class of Nondifferentiable Continuous Programming Problems,” European Journal of Pure and Applied Mathematics, 2010 (in press).

[10] 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.

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

[12] B. Mond and M. Schechter, “Nondifferentiable Symmetric Duality,” Bulletin of the Australian Mathematical Society, Vol. 53, 1996, pp. 177-188.