DUALITY IN MULTIOBJECTIVE PROGRAMMING UNDER GENERALIZED CONVEXITY

Abstract

The work being presented in the present thesis is devoted to the study of duality results for some multiobjective programming problems under generalized convexity assumptions. The chapterwise summary of the thesis is as follows: Chapter 1 consists of introduction to multiobjective programming, some defini-tions, notations 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 also presented. In mathematical programming, a pair of primal and dual problems is called symmetric if the dual of the dual is the primal problem. In Chapter 2, we con-sider two pairs of Wolfe and Mond-Weir type nondifferentiable multiobjective sym-metric dual programs where the objective function is optimized with respect to a closed convex cone K. Duality results are established under K-preinvexity/K-convexity/pseudoinvexity assumptions. Several known results have been deduced as special cases. Mangasarian [84] introduced the concept of second-order duality for nonlinear programs. This motivated several authors [1, 48, 57, 116, 128] in this field. In Chapter 3, we formulate a pair of Mond-Weir type second-order symmetric dual 11 problems and prove the duality relations under K-mbonvexity assumptions. Wolfe type duality has also been discussed. This work also serves to remove certain omis-sions in Mishra and Lai [861, wherein terms of different dimensions have been added in the objective functions and the constraints of Mond-Weir type symmetric dual problems. Chapter 4 deals with the second-order symmetric dual multiobjective programs over arbitrary cones, where every objective and constraint function contains a term involving a support function of a compact convex set. The first section contains intro-duction. In the next section a pair of Mond-Weir type second-order dual is formulated and duality theorems are proved under second-order K-F-convexity / K-r1-bonvexity assumptions. Self duality and special cases have been discussed in the last two sec-tions of the chapter. In Chapter 5, we consider Wolfe and Mond-Weir type multiobjective second-order symmetric dual programs over arbitrary cones in which the objective function is optimized with respect to an arbitrary closed convex cone K. For these dual pairs, duality results are established under K-preinvexity / K-F-convexity / pseudoinvexity assumptions. The dual models discussed here involve two functions f : S1 x S2 Rk and g : S1 x S2 ---+ Rr while the second-order symmetric dual problems considered in Chapter 3 involve the function f only. The mathematical programming problem in which the objective function is the ratio of two functions is called a fractional programming problem. In general it is non convex and so the Karush-Kuhn-Tucker optimality conditions are only necessary. In 111 Chapter 6, a new class of higher-order (V, a, p, 8)-invex functions is introduced. We first obtain conditions under which a fractional function is higher-order (V, a, p, 0)- invex. We then consider a multiobjective fractional programming problem and derive sufficient optimality conditions for its efficient solution. Based upon aforesaid func-tions duality results are derived for Schaible type and generalized dual programs. Mond and Hanson [93] introduced the concept of duality in variational problems. In Chapter 7, we first modify a converse duality theorem (Theorem 3.3 in [58]) for a second-order dual of a scalar variational problem. We then consider its multiob-jective analogue, and obtain necessary optimality conditions and duality relations for efficient solutions. At the end, the static case of our problems has also been discussed. In Chapter 8, certain omissions have been pointed out in some papers on sym-metric duality in multiobjective programming. Corrective measures have also been discussed.