OPTIMALITY CONDITIONS AND DUALITY. IN MATHEMATICAL PROGRAMMING

Abstract

The work being presented in the thesis is devoted to the study of optimality conditions and duality for some mathematical programming problems. The results obtained in the thesis are organised as follow : Chapter I consists of introduction to nonlinear programming, some definitions and prerequisites for the present work. A brief account of the related studies made by various authors in the field and a summary of the thesis are presented. In Chapter II, we consider Wolfe and Mond-Weir type second order symmetric dual programs and establish duality theorems under ri 1-bonvexity/3h -boncavity and ii-pseudobonvexity/i, -pseudoboncavity of the kernel function K(x,y). These duality results are then used to study second order minimax mixed integer dual problems. Self duality theorems have also been obtained. Finally, it is observed that for a particular function K both pairs of symmetric dual programs are reduced to the general nonlinear problem and its Wolfe type second order dual introduced by Mangasarian [64]. In Chapter III, Wolfe and Mond-Weir type symmetric dual programs are formulated for nondifferentiable minimax mixed integer mathematical programming problems and symmetric duality theorems are obtained. Self duality for these pairs has also been discussed. These results generalize the work of Chandra et al. [17], Mond and Weir [78] and Mond [70]. Chapter IV deals with a multiobjective nonlinear fractional programming problem. Fritz John and Kuhn-Tucker type necessary conditions are derived and duality theorems are established for Jagannathan and Mond-Weir type duals relating the efficient solutions of the primal and dual problems. Various duality results including those of Egudo [35] and Weir and Mond [107] are obtained as particular cases. In Chapter V, a nondifferentiable multiobjective programming problem is considered and it is shown that the Fritz John type necessary conditions for an efficient solution obtained in Kanniappan [56] are in fact both necessary and sufficient (under certain convexity assumption) for a weak efficient solution . Kuhn-Tucker type necessary and sufficient conditions are also derived for a weak efficient solution. Moreover, we show that the Kuhn-Tucker type sufficient conditions of Gulati and Talaat [45] are also necessary for a properly efficient solution and use these necessary conditions to establish a strong duality theorem for a Mond-Weir type dual. Finally, a converse duality theorem is discussed for a special case of nondifferentiable multiobjective program, where subgradients can be computed explicitly. Bector et al. [10] derived Fritz John and Kuhn-Tucker type necessary and sufficient conditions and discussed Mond-Weir type duality for efficient solutions of a nondifferentiable multiobjective fractional programming problem. In Chapter VI, we generalize their work. The Fritz John type necessary conditions for efficiency in Bector et al.[10] are shown to be both necessary and sufficient for weak efficiency. We also derive Kuhn-Tucker type necessary and sufficient conditions for weak efficient solutions and show that the Kuhn-Tucker type necessary conditions in Bector et al.[10] are sufficient for a properly efficient solution under a certain boundedness condition. This last result gives conditions under which an efficient solution is properly efficient. An example is also discussed to illustrate this result. Moreover, our duality relations are for properly efficient solutions. Chapter VII deals with Wolfe and Mond-Weir type symmetric dual variational problems. Duality theorems are established under convexity/concavity and pseudoconvexity/pseudoconcavity assumptions. Self duality is also discussed. Our duality theorems can be viewed as dynamic generalization of the corresponding (static) symmetric and self duality theorems of minimax nonlinear mixed integer programming problems of Balas [2] and Kumar et al. [61.1.