OPTIMALITY AND DUALITY IN MULTIOBJECTIVE PROGRAMMING UNDER GENERALIZED CONVEXITY

Abstract

The aim of the present thesis is to study optimality conditions and duality results for some multiobjective programming problems under generalized convex as-sumptions. We have extended the notion of convexity introduced by Hachitni and Aghezzaf [42] to new classes of generalized convexity. Also, an example is given to show the existence of such functions. The optimality conditions and duality results for some multiobjective programming problems are studied under these generalized convexity assumptions. An attempt is also made to correct certain omissions in two recent papers. The chapterwise summary of the thesis is as follows: Chapter 1 consists of introduction to differentiable multiobjective programming and definitions, 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. Recently Yang et al. [100] observed an inconsistency in an assumption and the conclusion of the Huard type converse duality theorem in [52] and established the modified proof for the same. In Chapter 2, we have obtained an alternative proof of the converse duality theorem under weaker assumptions. The inconsistency observed ii by Yang et al. [100] in [52] has also appeared in the converse duality theorem for the multiobjective problem studied in [83], for which a modified proof is presented in Section 2.3. In Chapter 3, we have considered the nonlinear multiobjective programming problem, where the objective and constraint functions are locally Lipschitz on an open set X C R. The chapter is divided into five sections. In Section 3.2 a new class of generalized (F, a, p, d)—type I functions is introduced for a nonsmooth multiobjective programming problem, which extends the work of Hachimi and Aghezzaf [42]. Based upon these generalized functions, in Section 3.3, we have obtained the sufficient op-timality conditions. In Sections 3.4 and 3.5 respectively, the duality results for Wolfe type and Mond-Weir type dual programs are established. Chapter 4 deals with a nonlinear multiobjective programming problem, where the objective and constraint functions are differentiable on an open set X C Rn. Along the lines of Hachimi and Aghezzaf [42] and Jeyakumar and Mond [58], we have introduced a new class of generalized convex functions named (F, a, p, — V--type I. As an example, we have considered the multiobjective programming problem: Minimize f (xi, x2) = (x2(11- — x2)ecc)s Xi, sine xi, x1 + cos x2) subject to (xi, x2) E X, gi = 7r 4 xi 5- 0, = —cos x2 0 where X = {(x1, x2) : 0 < x1 < 0 < x2 < 1-"} and shown that the concept of (F, a, p, d) — V—type I functions introduced in this section is new and generalized. Section 4.3 contains several sufficiency theorems for a feasible solution of the primal ill problem to be its efficient/weak efficient solution. Then a Mond-Weir type duality has been discussed in the next section. Also, some of the results have been verified using the above example. In Chapter 5, we have established the duality results for second order dual programs. Firstly, in Section 5.2, the concept of second order V-type I functions is defined and the duality results are established for a Mond-Weir type multiobjec-tive dual program. In the next section, as an extension of the functions introduced in Chapter 4, a new class of second order (F, a, p, d)-V-type I functions is defined. Based upon these functions weak, strong and strict converse duality theorems are derived for Mond-Weir type multiobjective dual program. In Chapter 6, we have considered a nondifferentiable multiobjective programming problem, where every component of the objective and constraint functions contains a term involving the support function of a compact convex set. The first two sections contain introduction and definitions. Then following Kanniappan [60], the neces-sary optimality conditions introduced by Husain et al. [53] for a nondifferentiable scalar programming problem are extended to a nondifferentiable multiobjective pro-gramming problem. Further, Wolfe type dual and a unified higher order dual are formulated and the duality theorems are established. Several known results can be deduced as special cases. In mathematical programming, a pair of primal and dual problems is called symmetric if the dual of the dual is the primal problem. However, the majority iv of dual formulations in nonlinear programming do not possess this property. Com-plex mathematical programming, the extension to complex variables and functions of mathematical programming was initiated by Levinson [70], where the duality theory for complex linear programming is given and the basic theorems of linear inequalities are extended to complex space. Bhatia [16] studied symmetric duality in complex space. Recently, Mishra and Rueda [80] extended the work of Chandra et al. [19] on symmetric duality to complex space. However the complex variables z and v have been restricted as z > 0 and v 0 respectively, which appear to have no meaning for complex vector variables z and v. In Chapter 7, Section 7.3 contains the correct Wolfe type symmetric dual models over general polyhedral cones and the duality re-sults are established for the same under generalized convexity assumptions. Next we have formulated the second order Wolfe type symmetric dual programs over general polyhedral cones in complex spaces and stated the duality results. Further, two ob-servations regarding the strong duality theorem have been made and finally, in the last section, we have stated weak duality theorems for Wolfe type nondifferentiable first and second order primal and dual programs