【作者】 龙宪军；

【导师】 彭建文；

【作者基本信息】 重庆师范大学 ， 运筹学与控制论， 2006， 硕士

【摘要】 凸性和广义凸性在数理经济、工程、管理科学以及在最优化理论中起着非常重要的作用。因此，对凸函数和广义凸函数的研究是数学规划中最重要的内容之一。本文主要对两类广义凸函数做了进一步的研究。首先，本文提出了一类新的广义凸函数，即半-B-预不变凸函数，这类函数是半预不变凸函数和B-凸函数的推广，因此半-B-预不变凸函数概念的提出有一定理论意义，本文从以下几个方面研究了这类广义凸函数：(1)举例说明了这类函数的存在性，并给出反例说明半-B-预不变凸函数是半预不变凸函数和B-凸函数的真推广；(2)给出了半-B-预不变凸函数的一些性质；(3)讨论了半-B-预不变凸函数在极小化问题中的应用。本文研究的第二类广义凸函数是由Bector、Duneja和Gupta[22]引入的一致凸函数，它既是不变凸函数的推广，又是v-不变凸函数的推广，作者当时只考虑了目标函数是一致凸函数的多目标规划的最蚀陛条件和对偶定理.本文考虑了关于一致凸函数的多目标分式规划问题，获得了弱对偶定理、强对偶定理和严格逆对偶定理。而且，本文利用Clarke广义方向导数针对Lipschitz函数在原来一致凸函数概念的基础上定义了不可微的一致凸函数，并利用这类新凸性，我们研究了非光滑多目标分式规划，获得了广义Karush-Kuhn-Tucher最优性条件；弱对偶定理、强对偶定理和严格逆对偶定理。 另一方面，与凸性紧密相关的的一个概念是单调性。众所周知，一个实值函数是凸函数等价于对应的梯度函数是单调的。而单调性在研究变分不等式、变分包含和相补问题解的存在性和灵敏性中起着非常重要的作用。因此，本文最后一部分正是利用广义单调性来讨论了一类完全广义强非线性隐拟变分包含问题解的灵敏性．更多还原

【Abstract】 Convexity and generalized convexity play a central role in mathematical economics, engineering, management science, and optimization theory. Therefore, the research on convexity and generalized convexity is one of the most important aspects in mathematical programming. In this paper, we mainly make fruther research about two classes of generalized convex functions. First, a class of functions called semi-B-preinvex functions, which is a generalization of the semipreinvex functions and B-vex function , is introduced. Then the introduction of semi-B-preinvex functions has many theory significance. In the paper, we study this class of generalized convex functions from the following aspects: (l)give examples to show that there exist functions which are semi-B-preinvex functions but are neither semipreinvex nor B-vex; (2)obtain some basic properties of semi-B-preinvex functions; (3)some results for the extremum problem which the objective function is semi-B-preinvex functions are presented. Univex functions is the second class generalized functions we considered in this paper, which was presented by Bector, Duneja and Gupta [22] and which is a generalization of invex functions and v-invex functions. Optimization and duality results are also obtained for a nonlinear multiobjective programming problem in [22]. In this paper, we are considered the multiobjective fractional programming problem. Some duality theorems for multiobjective fractional programming problems with univex functions are obtained. Furthermor, we consider their nondiferentiable situation, we define nonsmooth univex functions for Lipschitz functions by using Clarke generalized directional derivative and study nonsmooth multiobjective fractional programming with the new convexity. We establish generalized Karush-Kuhn-Tucker necessary and sufficient optimality condition and prove weak, strong and strict converse duality theorems for nonsmooth multiobjective fractional programming problems containing univex functions.On the other hand, a concept closely related to the conveity is the monotonicity. It is well known that the convexity of a real-valued function is equivalent to the monotonicity of the corresponding gradient function. It is worth noting that monotonicity played a very important role in studying the existence and the sensitivity analysis of solutions for variational inequality, variational inclusions and complementarity problems. Theorefore, the last section of this paper we study the sensitivity of the solution of completely generalized strongly nonlinear implicit quasi-variational更多还原