SYMMETRIC DUALITY IN MATHEMATICAL PROGRAMMING PROBLEMS

Abstract

In mathematical programming, a pair of primal and dual problems is called symmetric if the dual of the dual is the primal problem. The present thesis is devoted to the study of symmetric duality for some mathematical programming problems under generalized convexity assumptions. The thesis is divided into eight chapters. The chapter wise summary of the thesis is as follows: Chapter 1 consists of introduction to mathematical programming, some de nitions, notations and prerequisites for the present work. A brief account of the related studies made by various authors in the eld and a summary of the thesis are also presented. Bector and Chandra [19] discussed second-order symmetric duality in mathematical programming under second-order pseudo-convexity/concavity assumptions. Chen [35, 36] studied higher-order symmetric duality for scalar and multiobjective nondi erentiable programming problems by introducing higher-order F-convexity. Mond-Weir type duality has been discussed in both these papers. Chapter 2 is devoted to the study of higher-order symmetric duality. We formulate Wolfe and Mond-Weir type higher-order symmetric dual programs and discuss duality relations under higher order invexity/pesudoinvexity assumptions. In continuation, we also introduce a pair of i ii mixed type higher-order symmetric dual problems. Wolfe and Mond-Weir type duals are special cases of these models. The weak, strong and converse duality theorems are established for these programs. Our study extends some of the known results. Bector et al. [20] and Yang et al. [121] studied mixed symmetric duality in mathematical programming. Recently, Agarwal et al. [1] and Gupta and Kailey [52] studied second order mixed symmetric duality. In Chapter 3, we consider a di erent pair of mixed symmetric dual problems and established usual duality theorems under second order F􀀀convexity assumptions. The mathematical programming problem in which the objective function is the ratio of two functions is called a fractional programming problem. In general, such problems are nonconvex. A pair of symmetric dual second-order fractional programming problems has been modeled in Chapter 4 and appropriate duality theorems have been established under 1-bonvexity / 2􀀀boncavity assumptions. These duality results are then used to study second-order minimax mixed integer dual problems. Our study extends some of the known results in [22, 29, 34, 44, 82] Chapter 5 deals with Wolfe and Mond-Weir type nondi erentiable multiobjective second-order symmetric dual programs. For these dual pairs, duality results are established under -invexity/ -pseudoinvexity assumptions. The dual models discussed here involve two di erent kernel functions f : S1 S2 ! Rk and g : S1 S2 ! Rr, while the second-order symmetric dual problems considered in Chapter 6 involve the function f only. In Chapter 6, we consider a pair of second-order Wolfe type nondi erentiable iii multiobjective symmetric dual programs, where the objective function is optimized with respect to a closed convex cone. Duality results are established under K- - bonvexity assumptions. Several known results have been deduced as special cases. Recently, Nahak and Padhan [103] have presented higher order symmetric dual programs for multiobjective problems. In the literature, strong and converse duality theorems have been established assuming conditions on known quantities. However, in strong and converse duality theorems in [103], an assumption involves the unknown lagrange multiplier 2 Rk forWolfe type symmetric duals, and the lagrange multipliers 2 Rk and for Mond-Weir type symmetric duals. In Chapter 7, a pair ofWolfe type higher-order symmetric dual multiobjective programs is formulated. Strong and converse duality theorems are established under invexity assumptions. Duality relations for Mond-Weir type dual models have also been obtained under pseudoinvexity assumptions. In this chapter we establish these results under the assumptions on the lines of [33, 46, 52, 57, 68] involving known quantities. Mond and Hanson [92] introduced the concept of duality in variational problems. Chapter 8 contains second-order symmetric dual multiobjective variational problems. We established duality theorems for this pair under the assumptions of -bonvexity/ -pseudobonvexity. At the end of this chapter, the static case of our problems has also been stated.