The Research Of The Theory On Generalized Invex Functions And Multiobjective Programming

The organization and content of this thesis can be summarized as follow: In chapter 2, on one hand, the relation between the strongly invex(pseudoinvex) functions and the strongly invariant monotone (pseudomonotone) of their gradients is discussed. It is obtained that a function f(x) is pseudoinvex with respect to a vector function η iff it is prequasiinvex with respect to the same vector function η when it is satisfied some conditions. On the other hand, we introduce the general subgradient, and then give a quite effcient criteria of the global minimums for preinvex functions satisfied locally Lipschitz condition. In chapter 3, firstly, an equivalent multiobjec-tive programming problem named (VP(η,θ,ρ)) is constructed by a modification of the objective function of (VP) with (α,ρ)-invex objective functions and (β, q)-invex constraint functions. Furthermore, optimal sufficient conditions for (VP) are discussed with the help of a (η,θ,ρ)-Lagrange function and a new type of saddle points. At the same time, another optimal sufficient conditions for a multiobjective fractional programming problem are given with generalized (α,ρ)-invex functions by Gordan theorem. In chapter 4, nondifferentiable vector functions of type I (NV-Type I), nondifferentiable vector functions of type II (NV-Type II) and generalized NV-Type I are defined by the general subgradient. Moreover, through condition C, the relations between preinvex functions and objective functions which are NV-Type I or NV-Type II are also disscussed. In the end, the optimal conditions and weak duality, strong duality, converse duality of the multiobjective programming with objective functions and restricted functions which are generalized NV-Type I are given.