Optimality And Duality For A Class Of Generalized Invex Functions

As is well known,optimization is the problem that people frequently en-countered in the engineering,scientific research and economic management fields. In practise,we often need to study multiobjective optimization problems under some constraint conditions.In recent decades.many mathematicians work for the studying of multiobjective optimization,constructing their dual problems,and obtaining the sufficient and necessary optimal conditions.Convexity is a basic mathematical concept and plays an important role in many mathematical problems. Particularly,the primary optimization theory is to discuss convex optimization problems. However,the limitations of convexity conditions are also very obvious,since a large number of practical problems are not satisfied with convexity conditions. Therefore,the relaxation of convexity conditions,the generalization of the convex function concept have both theoretical significance and practical application background. In recent years,generalized convex functions have been paid an increasing amount of attention. People put forward a number of generalized convex functions from different direction,and apply them to multi-objective optimization problem.In this paper,we will introduce generalized d-ρηθ-univex function,and discuss the relationships among d-invex function,d-univex function,d-ρηθ-invex function and d-ρηθ-univex function.We claim that d-ρηθ-univex function is the generalization of d-invex function,d-univex function,d-ρηθ- invex function by giving some examples. Further,we will discuss the following multiobjective programming problem for generalized d-ρηθ-univex functions (P)min f(x) s.t. g(x)≤0, x∈X. Where f:X→Rk,g:X→Rm,X(?)Rn is a nonempty subset.This paper is composed of four chapters and organized as follows:In chapter one,we first give some notions and definitions,and then briefly review the developmental background and studying progress.In chapter two,we introduce generalized d-ρηθ-univex function,and dis-cuss the relationships between d-ρηθ-univex function and the known concepts appeared in recent papers. Under generalized d-ρηθ-univexity assumption, we give the sufficient optimality conditions for the existence of the weak Pareto efficient solution of the problem (P).In chapter three,under certain conditions,the results on the weak,strong and converse duality of Mond-Weir type duality(MWD),and on the weak and strong duality assertions of the generalized Mond-Weir type duality(GMWD)of the problem (P) are obtained.In chapter four,under generalized d-ρηθ-univex condition,we construct other two dual types of the primal problem(P):Wolfe type duality and mixed type duality,and obtain the results on the weak,strong and converse duality of the two types duality. Our results generalize the relative results in some recent papers.