摘要
本文研究广义凸性及其在极值问题、对偶问题、Hahn-Banach定理和向量拟平衡系统问题等最优化问题中的一些应用。主要工作如下:
在第二章里,我们得到了严格预不变凸函数的两个性质,这些性质包括与中间点严格预不变凸性和预不变凸性有关的一个充分条件以及与半严格预不变凸性和中间点严格预不变凸性有关的一个充要条件。我们证明了两个预不变凸函数的比是不变凸函数,因此,我们对Yang、Yang和Teo在文献中提出的公开问题作出了肯定的回答。
在第三章里,我们首先得到了严格B-预不变凸函数的一个充分条件,然后给出了严格B-预不变凸函数的一些性质,最后讨论了严格B-预不变凸函数在极值问题中的应用。
在第四章里,我们纠正了文献的定理4.6或定理4.7中的错误,并用η关于第一变元是仿射的和η是斜对称的这两个条件代替η满足条件C,得到了(严格)伪不变单调性和拟不变单调性的新的必要条件。
在第五章里,我们首先引入了向量值映射的D-预不变凸性、D-半严格预不变凸性和D-严格预不变凸性概念,其次我们在*-半连续和*-下半连续条件下给出了D-预不变凸映射的一些性质,最后讨论了D-预不变凸性、D-半严格预不变凸性和D-严格预不变凸性的相互关系。
在第六章里,我们引入了向量值映射的D-预不变真拟凸性、D-严格预不变真拟凸性和D-半严格预不变真拟凸性概念,分别利用向量值映射的上D-半连续和下D-半连续概念,获得了D-预不变真拟凸向量值映射的等价结果。另外,我们还讨论了向量值映射的D-预不变真拟凸性、D-严格预不变真拟凸性和D-半严格预不变真拟凸性的关系,并证明了在一定条件下,向量优化问题的局部弱有效解一定是其全局弱有效解。
在第七章里,我们构造了两类不可微多目标规划问题的广义对偶模型,并建立了这些模型的弱对偶定理。
在第八章里,我们首先得到了几个新结果,它们将数量或向量情形的Hahn-Banach定理推广到集值情形。然后,我们证明了集值映射的Borwein-强次梯度和
In this thesis, the generalized convexity and their applications in optimization problems such as extremum problems, dual problems, Hahn-Banach theorems and system of vector quasi-equilibrium problems etc. are researched. The main research works are as follows:In Chapter 2, two properties of strictly preinvex functions are obtained. These properties include a sufficient condition in terms of intermediate-point strict prein-vexity and preinvexity and a necessary and sufficient condition in terms of the semistrict preinvexity and intermediate-point strict preinvexity. we also show that the ratio of preinvex functions is invex. Hence, we give a positive answer to the open question which was proposed by Yang, Yang and Teo in [1].In Chapter 3, a sufficient condition of the strictly B-preinvex function is firstly obtained. Then some properties of the strictly B-preinvex function are shown. Finally, some results for the extremum problem which the objective function is strictly B-preinvex are presented.In Chapter 4, the errors of Theorem 4.6 and Theorem 4.7 in [2] will be correted. And some necessary conditions of (strictly) pseudo-invex monotonicity and quasi-invex monotonicity are established with the condition that is affine in the first argument and skew instead of Condition C.In Chapter 5, the definitions of D-preinvexity, D-semistrict preinvexity and D-strict preinvexity for vector-valued maps are firstly introduced. Then, under the conditions of *-upper semi-continuity and *-lower semi-continuity, some properties of D-preinvexity are given and the interrelations among D-preinvexity, D-semistrict preinvexity and D-strict preinvexity are discussed.
In Chapter 6, the definitions such as D-proper prequasiinvexity, D-properly strict prequasiinvexity, D-properly semistrict prequasiinvexity for vector-valued maps are introduced, and under the lower D-Semi-continuous condition or the upper D-Semi-continuous condition, respectively, some equivalent propositions of D-proper prequasiinvexity for vector-valued maps are obtained, and the relationships of D-proper prequasiinvexity, D-properly semistrict prequasiinvexity and D-properly strict prequasiinvexity are discussed, under some conditions the local weak efficient solution of (VP) must be the global weak efficient solution of (VP) is proved.In Chapter 7, two new dual models of the nonsmooth multiobjective programming are constructed and two weak duality theorems for these models are derived.In Chapter 8, Some new results which generalize the Hahn-Banach theorems from scalar or vector-valued case to set-valued case are firstly obtained. Then, the existence of the Borwein-strong subgradient and the Yang-weak subgradient for set-valued maps are also proven. Finally, A new Lagrange multiplier theorem and a new Sandwich theorem for set-valued maps are also presented.In Chapter 9, the notion of affinelike set-valued maps is introduced and some properties of these maps are presented. Then a new Hahn-Banach extension theorem with a D-convex set-valued map dominated by an affinelike set-valued map whichcontains the main results in charter 8 as special cases is obtained.In Chapter 10, we introduce the definition of Di-0-partiaHy diagonal quasicon-vexity which is a generalization of D-quasiconvexity. We also introduce a new system of vector quasi-equilibrium problems and prove its existence of a solution. As applications, some existence results of weak pareto equilibrium for both constrained multicriteria games and multicriteria games without constrained correspondences are also shown.
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