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Continuous Programming Containing Support Functions
I. Husain,Z. Jabeen 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.1
In this paper, we derive necessary optimality conditions for a continuous programming problem in which both objective and constraint functions contain support functions and is, therefore, nondifferentiable. It is shown that under generalized invexity of functionals, Karush-Kuhn- Tucker type optimality conditions for the continuous programming problem are also sufficient. Using these optimality conditions, we construct dual problems of both Wolfe and Mond-Weir types and validate appropriate duality theorems under invexity and generalized invexity. A mixed type dual is also proposed and duality results are validated under generalized invexity. A special case which often occurs in mathematical programming is that in which the support function is the square root of a positive semidefinite quadratic form. Further, it is also pointed out that our results can be considered as dynamic generalizations of those of (static) nonlinear programming with support functions recently incorporated in the literature. In this paper, we derive necessary optimality conditions for a continuous programming problem in which both objective and constraint functions contain support functions and is, therefore, nondifferentiable. It is shown that under generalized invexity of functionals, Karush-Kuhn- Tucker type optimality conditions for the continuous programming problem are also sufficient. Using these optimality conditions, we construct dual problems of both Wolfe and Mond-Weir types and validate appropriate duality theorems under invexity and generalized invexity. A mixed type dual is also proposed and duality results are validated under generalized invexity. A special case which often occurs in mathematical programming is that in which the support function is the square root of a positive semidefinite quadratic form. Further, it is also pointed out that our results can be considered as dynamic generalizations of those of (static) nonlinear programming with support functions recently incorporated in the literature.
Mixed type second-order duality with support function
I. Husain,A. Ahmed,Mashoob Masoodi 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5
Mixed type second order dual to the non-differentiable problem containing support functions is formulated and duality theorems are proved under generalized second order convexity conditions. It is pointed out that the mixed type duality results already reported in the literature are the special cases of our results. Mixed type second order dual to the non-differentiable problem containing support functions is formulated and duality theorems are proved under generalized second order convexity conditions. It is pointed out that the mixed type duality results already reported in the literature are the special cases of our results.
Mixed Type Duality for Control Problems with Generalized Invexity
I. Husain,A. Ahmed 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.5
A mixed type dual to the control problem in order to unify Wolfe and Mond-Weir type dual control problem is presented in various duality results are validated and the generalized invexity assumptions. It is pointed out that our results can be extended to the control problems with free boundary conditions. The duality results for nonlinear programming problems already existing in the literature are deduced as special cases of our results..
On fractional programming containing support functions
I. Husain,Z. Jabeen 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.18 No.1-2
Optimality conditions are derived for a nonlinear fractional program in which a support function appears in the numerator and denominator of the objective function as well as in each constraint function. As an application of these optimality conditions, a dual to this program is formulated and various duality results are established under generalized convexity. Several known results are deduced as special cases.
MULTIOBJECTIVE CONTINUOUS PROGRAMMING CONTAINING SUPPORT FUNCTIONS
Husain, I.,Ahmed, A.,Rumana, G. Mattoo The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.3
Wolfe and Mond-Weir type dual to a nondifferentiable continuous programming containing support functions are formulated and duality is investigated for these two dual models under invexity and generalized invexity. A close relationship of our duality results with those of nondifferentiable nonlinear programming problem is also pointed out.
I. Husain,A. Ahmed,G. MATTOO 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.1
A multiobjective variational problem involving higher order derivatives is considered and Fritz-John and Karush-Kuhn-Tucker type optimality conditions for this problem are derived. As an application of Karush-Kuhn-Tucker optimality conditions, Wolfe type dual to this variational problem is constructed and various duality results are validated under generalized invexity. Some special cases are mentioned and it is also pointed out that our results can be considered as a dynamic generalization of the already existing results in nonlinear programming. A multiobjective variational problem involving higher order derivatives is considered and Fritz-John and Karush-Kuhn-Tucker type optimality conditions for this problem are derived. As an application of Karush-Kuhn-Tucker optimality conditions, Wolfe type dual to this variational problem is constructed and various duality results are validated under generalized invexity. Some special cases are mentioned and it is also pointed out that our results can be considered as a dynamic generalization of the already existing results in nonlinear programming.
ON FRACTIONAL PROGRAMMING CONTAINING SUPPORT FUNCTIONS
HUSAIN, I.,JABEEN, Z. 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.18 No.1
Optimality conditions are derived for a nonlinear fractional program in which a support function appears in the numerator and denominator of the objective function as well as in each constraint function. As an application of these optimality conditions, a dual to this program is formulated and various duality results are established under generalized convexity. Several known results are deduced as special cases.
MIXED TYPE DUALITY FOR CONTROL PROBLEMS WITH GENERALIZED INVEXITY
Husain, I.,Ahmed, A.,Ahmad, B. Korean Society of Computational and Applied Mathem 2008 Journal of applied mathematics & informatics Vol.26 No.5
A mixed type dual to the control problem in order to unify Wolfe and Mond-Weir type dual control problem is presented in various duality results are validated and the generalized invexity assumptions. It is pointed out that our results can be extended to the control problems with free boundary conditions. The duality results for nonlinear programming problems already existing in the literature are deduced as special cases of our results.
Nondifferentiable second order self and symmetric dual multiobjective programs
I. Husain,A.Ahmed,Mashoob Masoodi 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.3-4
In this paper, we construct a pair of Wolfe type second order symmetric dual problems, in which each component of the objective function contains support function and is, therefore, nondifferentiable. For this problem, we validateweak, strong and converse duality theorems under bonvexity – boncavity assumptions. A second order self duality theorem is also proved under additional appropriate conditions.