Duality in Mathematical Programming Under Generalized Convexity

Abstract

The work being presented in the present thesis is devoted to the study of duality results for some mathematical programming problems under generalized convexity assumptions. The chapterwise summary of the thesis is as follows: Chapter 1 is introductory and consist of nonlinear and multiobjective programming problems, definitions, notations and prerequisites of 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 Chapter 2, we have considered Wolfe type second-order multiobjective symmetric dual programs involving nondifferentiable functions and appropriate duality theorems are established using the notion of second-order F-convexity assumptions. Moreover, an example has been given which is second-order F-convex but not convex. Further, these symmetric dual programs are generalized over arbitrary cones and usual duality results are obtained under second-order (F, , , d)-convexity assumptions. A non-trivial example which shows that second-order (F, , , d)-convex functions are generalization of second-order F-convex functions has also been exemplified. iii iv In Chapter 3, a new pair of second-order multiobjective symmetric dual programs in which the objective function is optimized with respect to an arbitrary closed convex cone is formulated and appropriate duality relations are then obtained under K- -bonvexity assumptions. We identify a function lying exclusively in the class of K- -bonvex and not in class of invex function already existing in literature. Self duality for this pair is also obtained by assuming the functions involved to be skewsymmetric. Further, we have considered a pair of Mond-Weir type nondifferentiable multiobjective second-order symmetric dual programs over arbitrary cones, where each of the objective function contains a square root term with positive semidefinite matrix in Rn×n. Weak, strong and converse duality results are then established under K- -bonvexity/second-order K-F-convexity assumptions. In Chapter 4, we have established duality relations for a pair of second-order mixed symmetric dual programs involving nondifferentiable functions under secondorder F-convexity/pseudoconvexity assumptions. Next, we have considered a pair of mixed second-order symmetric dual programs over cones and obtained duality results under second-order (F, ) convexity/pseudoconvexity assumptions. This mixed formulation unifies two second-order symmetric dual formulations exist in the literature. Several known results [39, 55, 57, 70] are obtained as special cases. In Chapter 5, we have formulated a pair of second-order multiobjective mixed symmetric dual programs over arbitrary cones and obtained appropriate duality v results under second-order invexity/pseudoinvexity assumptions. Further, we construct a pair of multiobjective second-order mixed nondifferentiable symmetric dual programs involving the square root of a positive semidefinite quadratic function, (xTBx) 1 2 . The usual duality results are then established using the notion of secondorder F-convexity/pseudoconvexity assumptions. Agarwal et al. [3] extended the results of Chen [43] over arbitrary cones and proved appropriate duality relations under higher-order K-F-convexity assumptions. Mond-Weir type duality has been discussed in both the papers. In Chapter 6, we have studied higher-order Wolfe type multiobjective symmetric dual programs over arbitrary cones and the duality results are then established under higher-order (F, , , d)-convexity/pseudo-convexity assumptions. We have also illustrated a nontrivial example of function lying in the class of higher-order K-(F, , , d)-convex but not in class of higher-order K-F-convex. Further, we consider the higher-order multiobjective symmetric nondifferentiable dual programs in which the objective function is optimized with respect to an arbitrary closed convex cone and proved duality theorems under higher-order-K-(F, , , d)-convexity assumptions. In Chapter 7, motivated by Lai et al. [85], Lai and Lee [84] and Antczak [19], we have discussed sufficient optimality conditions and duality theorems for a nondifferentiable minimax fractional programming problem with B-(p, r)-invexity. An example which is B-(1, 1)-invex but not (p, r)-invex is exemplified. We also illustrate another example which is (−1, 1)-invex but not convex. vi In Chapter 8, we have formulated a pair of multiobjective fractional variational symmetric dual problems for a class of nondifferentiable functions over arbitrary cones and achieved duality results under generalized (F, , , d)-convexity assumptions. A self duality theorem is also obtained by assuming the functions involved to be skewsymmetric. At the last, an Appendix A has been given, in which we establish a strong duality theorem for a pair of multiobjective second-order symmetric dual programs. This removes an omission in an earlier result in Yang et al. [