I. Husain

MULTIOBJECTIVE SYMMETRIC DUALITY WITH INVEXITY

VOL. 56 (1997) [25-36]MULTIOBJECTIVE SYMMETRIC DUALITY WITH INVEXITYT.R. GULATI, I. HUSAIN AND A. AHMEDUsual duality results are proved for Wolfe and Mond-Weir type multiobjective sym-metric dual problems without nonnegativity constraints under invexity/generalisedinvexity assumptions. Moreover, assuming the kernel function to be skew symmet-ric, the multiobjective problems are exhibited to be self duals.

Optimality conditions and duality for multiobjective control problems

Abstract Fritz John and Kuhn-Tucker type necessary optimality conditions for a Pareto optimal (efficient) solution of a multiobjective control problem are obtained by first reducing the multiobjective control problem to a system of single objective control problems, and then using already established optimality conditions. As an application of Kuhn-Tucker type optimality conditions, Wolfe and Mond-Weir type dual multiobjective control problems are formulated and usual duality results are established under invexity/generalized invexity, relating properly efficient solutions of the primal and dual problems. Wolfe and Mond-Weir type dual multiobjective control problems with free boundary conditions are also presented.

On multiobjective nonlinear programming with support functions

Abstract A multiobjective programming problem containing support functions is considered. Wolfe and Mond–Weir type vector duals to this problem are constructed and various duality results are validated under invexity and generalized invexity assumptions. Special cases are generated from these results.

Multiobjective Duality in Variational Problems with Higher Order Derivatives

A multiobjective variational problem involving higher order derivatives is considered and optimality condi-tions for this problem are derived. A Mond-Weir type dual to this 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.

Sufficiency and duality in control problems with generalized invexity

Abstract Sufficient optimality criteria are derived for a control problem under generalized invexity. A Mond-Weir type dual to the control problem is proposed and various duality theorems are validated under generalized invexity assumptions on functionals appearing in the problems. It is pointed out that these results can be applied to the control problem with free boundary conditions and have linkage with results for nonlinear programming problems in the presence of inequality and equality constraints already established in the literature.

Generalized Second-Order Duality for a Class of Nondifferentiable Continuous Programming Problems

The research in this paper is organized in two parts. In the first part, a generalized second-order dual is formulated for a class of continuous programming problems with square roots of positive semi-definite quadratic forms, hence it is nondifferentiable. Under second-order pseudoinvexity and second-order quasi-invexity, various duality theorems are proved for this pair of dual nondifferentiable continuous programming problems. Lastly, it is pointed out that our duality results can be regarded as dynamic generalizations of those for a nondifferentiable nonlinear programming problem, already treated in the literature. In the second part, a generalized second-order dual is formulated for a continuous programming problem with support functions. Duality results similar to those of the first part are validated for this class of problems. Clearly the results of this part are more general in sense that a support function extends a square root of a positive semidefinite quadratic form.

On a Class of Nondifferentiable Multiobjective Continuous Programming Problems

Fritz John and Karush-Kuhn-Tucker type optimality conditions for a nondifferentiable multiobjective variational problem with equality and inequality constraints are obtained. Using Karush-Kuhn-Tucker type optimality conditions, a Wolfe type second-order nondifferentiable multiobjective dual variational problem is constructed. Various duality results for the pair of Wolfe type second-order dual variational problems are proved under second-order pseudoinvexity. A pair of Wolfe type dual variational problems having equality and inequality constraints with natural boundary values is also formulated to prove various duality results. Finally, the linkage between our results and those of their static counterparts existing in the literature is briefly outlined.

Constrained Vector-Valued Dynamic Game and Symmetric Duality forMultiobjective Variational Problems

A certain constrained vector-valued dynamic game is formulated and shown to be equivalent to a pair of multiobjective symmetric dual variational problems which have more general formulations than those studied earlier. A number of duality theorems, are established under suitable generalized convexity assumptions on the functionals. Selfduality reflecting symmetric dynamic games is investigated. The constrained vector-valued dynamic game is also regarded as equivalent to a pair of symmetric multiobjective dual variational problems with natural boundary conditions rather than fixed end points. Finally, it is indicated that our results can be considered as dynamic generalizations of those already existing in the literature. AMS-Mathematics Subject Classification: Primary 90C30, Secondary 90C11, 90C20, 90C26.

On Multiobjective Duality For Variational Problems

In this paper two types of duals are considered for a class of variational problems involving higher order derivatives. The duality results are derived without any use of optimality conditions. One set of results is based on Mond- Weir type dual that has the same objective functional as the primal problem but different constraints. The second set of results is based on a dual of an auxiliary primal with single objective function. Under various convexity and generalized convexity assumptions, duality relationships between primal and its various duals are established. Problems with natural boundary values are considered and the analogs of our results in nonlinear programming are also indicated.

ERRATUM TO THE PAPER \SUFFICIENCY AND DUALITY IN CONTROL PROBLEMS WITH GENERALIZED INVEXITY"

The following typsetting mistakes have been found in the printed version of the paper I. Husain, A. Ahmed and B. Ahmad “Sufficiency and duality in control problems with generalized invexity”, Journal of Applied Analysis, Vol. 14, No. 1 (2008), pp. 27–42. 1. Page 42, line 25 from above, there is “Bashir Ahmad” while it should be “Bilal Ahmad” 2. Page 42, line 30 from above, there is “e-mail: bashir qau@yahoo.com” while it should be “e-mail: bilal.jalali@yahoo.com”