OPTIMALITY CONDITIONS AND DUALITY IN MULTIPLE OBJECTIVE PROGRAMMING

Abstract

The present thesis is devoted to the study of necessary and sufficient conditions for efficient/properly efficient solutions and duality for some nonlinear multiobjective programming prob-lems. The results obtained in the thesis are organized as follow : Chapter 1 consists of introduction to differentiable/ n ond if feren ti able multi obj e ctive programming, some definitions and prerequisites for the present work. A brief account of the related studies made by various authors in this field and a summary of the the sis are presented. In Chapter 2, a nonlinear vector minimum problem is consi-dered. Kuhn-Tucker type conditions are derived which are necessary for an efficient solution and sufficient for a properly efficient solution. These results imply that under a constraint qualifica-tion_ every efficient solution of a convex vector minimum problem is properly efficient. Fritz John type necessary and sufficient conditions for an eff icient solution are also- obtained. An example is discussed to illustrate the results. Moreover, a dual is formulated and duality relations are established for properly efficient solutions. The dual problem is different than in Weir [95,96] and for single objective primal problem it reduces to the well known Wolfe dual. The converse duality theorem- proved here requires nonsingularity of the Hessian matrix instead of positive or negative definiteness assumed in. Weir [96]. Bector et al [6] discussed duality theorems for a Mond—Weir type dual of a pseudolinear multiobjective programming problem assuming the same proportional function q(x,u) for objectives and r(x,u) for constraint functions. They also needed a constraint qualification for the converse duality theorem. In Chapter 3, we establish duality results of Sector et al [6] by a different approach. We prove strong and converse duality theorems using Kuhn—Tucker and Fritz John type necessary conditions respectively. This approach weakens the pseudolinearity assumption on objective functions to pseudo—convexity and avoids the need of a constraint qualification for the converse duality theorem. These results are applied to formulate duals for multiobjective fractional programs with different denominators. Like Bector et al [6], the duality relations of Chapter 3 are for efficient solutions. In Chapter 4, we establish these relations for properly efficient solutions. To do so, we need to assume stronger convexity assumptions, but these are still weaker than in [6]. The strong duality- theorem provides a properly efficient solution of the dual, while in converse duality theorem a weak efficient solution of the _dual gives a properly efficient solution of the primal problem. The chapter ends with some applications in fractional programming. In Chapter 5, a nondifferentiable multiobjective programm-ing problem is considered and Fritz John and Kuhn-Tucker type sufficient conditions are derived for efficient and properly efficient solutions respectively. An example is discussed to demonstrate the results. Weak and strong duality theorems are established for Wolfe type dual. Also, a converse duality theorem is proved for special class of nondifferentiable programs where subgradients can be computed explicitly. Finally, a Mond-Weir type dual is presented and its duality relations with the primal problem are stated. Chapter 6 deals with a nondifferentiable fractional multi-objective programming problem. Fritz John and Kuhn-Tucker type necessary and sufficient conditions for an efficient solution are derived and duality results are discussed for a Mond-Weir type dual. A converse duality theorem has been established for a special class of nondifferentiable fractional multiobjective programming problems. In the end, we state duality relations for Wolfe type dual of the primal problem. The thesis ends with an appendix. Here, a unified sufficient optimality theorem is derived for scalar nonline ar programming problems in the presence of both equality and inequality cons-traints. This result subsumes the well known Kuhn-Tucker and Fritz John sufficient theorems discussed separately in the literature by several authors.