DUALITY IN NONLINEAR 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 convex as-sumptions. The chapterwise summary of the thesis is as follows: Chapter 3. consists of introduction to nonlinear programming problems, 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 Chapter 2, we have considered Wolfe type second-order nondifferentiable dual programs and appropriate duality theorems are established under 771---bonvexity /?72—boncavity assumptions. Our results relax the nonnegativity conditions in the problems studied by Yang et al. [109] and subsume the work in [4, 24, 46, 82]. These duality results are then used to study second-order minimax mixed integer dual prob-lems. Self duality theorem have also been obtained. An example has also been given to illustrate the results. Chen [28, 29] studied higher-order symmetric duality foiscalar and inultiobjective nondifferentiable programming problems by introducing higher-order F-convexity. Mond-Weir type duality has been discussed in both these papers. In Chapter 3, we have formulated Wolfe type higher-order nondifferentiable symmetric dual pro-grams and discussed duality relations between them. These duality results have been used to discuss symmetric minimax mixed integer dual problems. Our study extends some of the known results, in [4, 24, 37, 46, 47, 81, 88, 109]. Moreover, we have also stated two dual pairs whose duality relations can be obtained from the results derived in this chapter. In Chapter 4, we have established duality relations for Wolfe and Mond-Weir type second-order symmetric duals over arbitrary cones under ri—bonvexity /77—pesudobonvexity assumptions. This work removes several omissions in the defi-nitions, models and proofs in [39]. An example has also been given which validates the duality results obtained in this chapter. These duality results are then used to study second-order minimax mixed integer dual problems. Self duality theorems have also been obtained. In Chapter 5, we have considered Wolfe and Mond-Weir type higher-order sym-metric dual programs with cone constraints and obtained appropriate duality results under higher-order invexity/pseudoinvexity assumptions. Examples have also been given which illustrate that the results obtained generalize the earlier results in the literature. Moreover, in the strong duality theorems in [7, 45, 46, 82], the assumption that the matrix {Vyykp}y or {Vy2y2kfi}y2 is positive or negative definite and the con-clusion that 25 = 0 are inconsistent. The work obtained in this chapter also overcomes this deficiency. Mishra and Rueda [86], assuming F-convexity established duality theorems for Wolfe and Mond-Weir type first and second-order symmetric dual programs in com-plex space. Their four pairs of primal and dual problems contain the constraints z > 0 and v > 0, respectively which appear to have no meaning for complex vector variables z and v. Moreover, there are several other omissions in the assumptions and proofs in [86]. In Chapter 6, we have modelled correct Mond-Weir type primal and dual programs over general polyhedral cones and the duality results are established for the same under generalized convexity assumptions. We have also formulated the second-order Mond-Weir type symmetric dual programs over general polyhedral cones in complex spaces and stated the duality results. Finally, in the last section, we have stated weak duality theorems for Mond-Weir type nondifferentiable first and second-order primal and dual pairs. In Chapter 7, a new class of generalized higher-order (F, a, fi, p, d)-convex functions is introduced which generalizes the notion of (F, a, p, d)-convex functions introduced by Liang et al. [76] and second-order (F, a, p, d)-convex functions defined by Ahmad and Husain [8]. We have established an alternative theorem under this class of functions. Next, we have formulated a multiobjective fractional programming problem involving support function of a compact convex set in each term of the ob-jective function and obtained sufficient optimality conditions related to weakly and properly efficient solutions of this problem. Weak and strong duality theorems are iv also derived for Wolfe and Mond-Weir type multiobjective nondifferentiable fractional dual programs.