OPTIMALITY AND DUALITY IN MATHEMATICAL PROGRAMMING UNDER GENERALIZED CONVEXITY

Abstract

The thrust of the present thesis is to study optimality conditions and duality re-sults for single as well as multiobjective programming problems under generalized convexity assumptions. The thesis is divided into six chapters which are described below : Chapter 1 is introductory in nature. A brief review of the related studies made by various authors in this field and an overview of the thesis is presented. In Chapter 2, we have formulated Wolfe and Mond-Weir type second-order multiobjective symmetric dual problems over arbitrary cones. Weak, strong and converse duality theorems are established under 77-bonvexity/n-pseudobonvexity as-sumptions. Moreover, self-duality of the primal problems is shown assuming the functions involved to be skew symmetric. This work removes several omissions in definitions, models and proofs for Wolfe type problems studied in Mishra (2000a,b). Chapter 3 deals with a class of nondifferentiable multiobjective programming problems. This Chapter is divided into three sections. In Section 3.2, we have formulated Wolfe type second order symmetric dual models for a class of nondif-ferentiable multiobjective programs. Weak, strong and converse duality theorems are established under second order F-convexity assumptions. In Section 3.3, we have generalized the dual models of Section 3.2 over arbitrary cones, where the ob-jective function is optimized with respect to an arbitrary closed convex cone and constraints are defined via closed convex cones and their polars. Using the concept of weak efficiency with respect to a convex cone, weak, strong and converse duality theorems are studied under second-order K-F-convexity assumptions. Self duality and special cases of the primal problems considered in this chapter have also been discussed. Chapter 4 deals with a nondifferentiable multiobjective fractional program-ming problem (P 4.1). Reducing this problem to a system of scalar nonlinear problems and using known results of scalar programming, both Fritz John and Karush-Kuhn-Tucker type necessary conditions for an efficient solution of (P 4.1) are obtained. Assuming convexity of numerator, concavity of denominator of the objectives and quasiconvexity of the active constraints, sufficiency theorems for an efficient solution of (P 4.1) are also derived. Under a certain boundedness assump-tion, Karush-Kuhn-Tucker type sufficient conditions for a properly efficient solution of (P 4.1) are established. This leads to sufficient conditions for an efficient solution of (P 4.1) to be properly efficient_ In Chapter 5, we have considered a nondifferentiable multiobjective fractional programming problem (P 5.1), where every component of the objective function and the constraints contain a term involving the support function of a compact convex set. In Section 5.2, the concept of higher-order (V, a, p, O)-invex functions is defined. In Section 5.3, we first derive an important result giving higher-order (V, a, p, 0)-invexity of the ratio of two functions and then use it to obtain Karush-Kuhn-Tucker type sufficient optimality conditions for an efficient solution of (P 5.1). In the last section, using this concept of generalized invexity weak and strong duality theorems are established for Mond-Weir type dual of (P 5.1). In Chapter 6, we have introduced the concept of higher-order (F, a, 3, p, d)-convexity. Under this generalized convexity we have established sufficient optimality conditions for a nonlinear programming problem (P 6.1), in Section 6.2. In Section 6.3, weak and strong duality relations for Mond-Weir type dual program for (P 6.1) are obtained. These duality results have been applied to obtain duality relations for a fractional programming problem in Section 6.4. The last section contains optimality ii and duality results for the multiobjective analogue of the fractional programming problem of Section 6.4. iii