摘要凸性是函数的一种很重要的性质，它在优化中起着重要的作用，是数学中的一个重要研究分支，在函数的研究领域中占有十分重要的地位。到目前为止，关于凸函数的性质已有很多的研究，而且在应用方面也有很多研究。对函数凹凸性的研究，在数学分析的多个分支都有用处。尤其是在函数图形的描绘和不等式的推导方面，凸函数起着十分重要的作用。凸函数因为有着良好的性质，所以不仅在理论方面有广泛的应用，而且在实际中也有广泛的应用。但是，由于凸性的条件要求比较高，在实际中很多函数并不满足严格的凸性条件，因此，需要对函数的凸性进行推广，这不仅是理论上的需求，也是实际应用中的需要。本论文重点研究一类推广的凸函数——拟凸函数。拟凸函数是数学研究函数凸性中的重要内容。由于凸函数的要求较高，在数学规划和经济学的许多问题中，当凸性的条件太强时,凸函数就无法胜任接下来的研究工作，这个时候就需要使用拟凸性。拟凸函数的要求更加的宽泛，在许多凸函数无法涉及的领域，拟凸函数得到了很好的使用。而且凸函数的大多数结果能够推广到拟凸函数。因此，研究拟凸函数的性质显得很重要,拟凸函数方向上的研究已构成当今数学规划问题研究中的重要方向之一。50093
毕业论文关键词 拟凸函数 上半连续 水平集 开凸锥 严格拟凸
Title Properties and applications of the quasi-convexfunction
Abstract Convexity is a very important property of the functions.It plays an important role in theoptimization and has a very important position in the field of research function.So far,we havenot only gotten a lot of research about the nature of convex function,but also gotten lots ofachievements of practical application.And we research concave and convex nature offunction,because it is very useful to the mathematical Analysis.Convex-Function plays a veryimportant role in the mathematical Analysis especially in function’s graph-depicting and theproof of inequality.Convex-function has good properties.It not only has widely used theory,butalso has widely use in practical application.However,,in practice many functions do not meetstrict convexity condition due to the convexity of the conditions is relatively high.So,we needpromote the nature of the convex-function.It is not only a theoretical demand, but also thepractical application needs .This paper focuses on researching a class of extension of convex function -- quasi-convexfunctions. Quasi-convex function is an important content of mathematical function whichresearches Convexity of . Because of the high requirements of the convex functions. In manyproblems of mathematical programming and economics the convex function can not be qualifiedfor the next research work when the condition is too strong convexity,at this time we need to usequasi-convexity. Quasi-convex function requires weak than convex function. In many areaswhich the convex functions can not be involved, maybe the Quasi-convex function has beenwell used. And most of the results of convex function can be generalized to the quasi-convexfunction . Therefore, the study of the quasi-convex function’s nature is very important. The studyof Quasi-convex function direction constitutes one of the research today mathematicalprogramming problem is an important direction.
Keywords Quasi convex function;Upper Semi-continuous Mappings; Level Set;Open convex cone;Strictly quasi-convex.

目次

1绪论1

2凸函数的预备知识5

2.1凸函数的定义和几何意义7

2.2凸函数与Jensen不等式8

2.3凸函数与Hadamard不等式10

3拟凸函数12

3.1拟凸性和单调性14

3.2连续性15

3.3可微性16

3.4拟凸函数与Hadamard不等式18

3.5拟凸函数与土凸函数的区别19 拟凸函数的性质及应用:http://www.youerw.com/shuxue/lunwen_53303.html