E-Convex Sets, E-Convex Functions And E-Convex Programming In Linear Spaces

In this thesis, some topics of E-convex sets, E-functions, E-programming are discussed in ordered linear spaces. Firstly, E-convex sets are defined in real linear spaces and some its properties are discussed. The properties of real E-convex functions ,semi-E-convex functions, E-quasiconvex functions defined on a E-convex set in the real linear spaces and the relations between E-quasiconvex functions(resp. strictly E-quasiconvex functions ) and quasiconvex functions (resp. strictly quasiconvex functions)are discussed. The two results with regard to E-convex functions in paper [8] are generalized in the real linear spaces. Those are prime in Chapter One. In Chapter Two, the vector-valued E-convex functions, semi-E-convex functions on a E-convex set are defined. All functions in this chapter and Chapter Three are these two functions. The alternative theorem is proved under the assumption of E-convex functions and semi-E-convex functions by means of the separate theorem of convex sets in the linear spaces. Under the assumption of E-convexity and semi-E-convexity, the optimality conditions, Lagrange multipliers theorems, saddle points theorems are established, and these are primary in this thesis. In Chapter Three, the differentiable vector optimization problems with relation to E-convexity and semi-E-convexity under the assumptions of Gateaux differentiable and Frechet differentiable are discussed.