广义均衡问题的研究
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摘要
均衡是一个研究许多实际生活现象中某些系统的一个核心概念,这包含了从经济、网络到力学等许多领域.均衡在现实中的应用研究也促进了不动点和最优化理论的发展.抽象变分不等式是表示这些系统的一个简单而自然的形式,它的另外一种表述就是著名的Ky Fan极大极小不等式,而均衡问题模型就是这两者的推广.均衡问题的数学模型不但为最优化、不动点理论、变分不等式和互补问题提供一个统一的形式,它还包含了其它一些重要的数学模型,如半变分不等式、Nash均衡等.向量均衡问题是均衡问题的一种重要推广,它包含了通常的多目标优化和向量变分不等式.除此之外,均衡问题还被推广为广义均衡问题、均衡系统问题、拟均衡问题和类均衡问题等,这些与最优化、控制理论、博弈论、工程及力学中的一些非线性分析问题联系密切,有着广泛的应用背景.
本文主要研究向量均衡问题、均衡系统问题和广义均衡问题解的存在性,讨论其求解算法.具体如下：
1.基于Gerstewitz非线性标量化函数和变动控制结构下的非线性标量函数,引入了序拓扑向量空间中集合的非线性标量化函数概念,讨论了其性质.对拓扑空间上集值映射的连续性做了推广,引入了集值映射的类上半连续概念,讨论了它和集值映射上半连续的关系.对著名的Ekland变分原理进行了推广,得到了集值映射形式的广义Ekland变分原理,它包含了向量形式的Ekland变分原理.
2.研究了拓扑向量空间上向量均衡问题解的存在性,建立了欧氏空间上向量均衡问题的迭代算法,用非线性标量化方法证明了该算法的收敛性.该算法是对数量均衡问题的Alfredo投影迭代算法的向量形式推广.对Tada提出的Hilbert空间上数量均衡问题的算法进行了推广,得到了Hilbert空间上向量均衡问题的两个算法.用非线性标量化方法证明了其中一个算法强收敛,而另一个算法是弱收敛的.
3.研究度量空间上的向量均衡系统问题.利用集值映射型广义Ekland变分原理,证明了向量均衡系统新的解的存在性结论.该结论只要求向量均衡系统中函数的满足某种连续性,而无需均衡系统的有效域满足任何凸性,也不要求函数满足单调性及广义凸性条件.
4.对Moudafi的粘性迭代方法和显性粘性迭代方法进行了推广,构造了一个混合迭代方案,用于求解Hilbert空间上一个混合均衡系统问题和无限个非扩张映射的不动点问题的公共解,作为推论得到了Hilbert空间上均衡系统问题和均衡问题的求解算法.
5.研究了度量空间上的集值映射型广义向量均衡问题,用集值映射型广义Ekland变分原理建立了其解的存在性定理.构造了欧氏空间上具有变动控制结构的广义向量均衡问题的一个投影迭代算法,它是Alfredo数量均衡问题投影迭代算法的一个推广
6.在广义凸空间上定义了一类新的广义均衡问题,研究了其解的存在性,利用不动点定理证明了满足集值映射类上半连续性条件的解的存在性定理.作为该定理的推论,推导出了均衡问题、向量均衡问题和广义向量拟均衡问题解的存在性的一些新结果.
Equilibrium is a central concept to study some systems governing many real-life phenomena in fields ranging from economics and networks to mechanics. Andthe study of equilibrium in real-world applications have promoted the develop-ment of fixed point theory and optimization. The abstract variational inequality isa simple and natural formulation for those systems. Ky Fan’s minimax inequalityis another statement. And the equilibrium problem formulation is a generaliza-tion for them. The equilibrium problem formulation not only proposed a uni-fied model including optimization, fixed point theorems, variational inequalitiesand complementarity problems, but also contained other important mathemati-cal models, such as best approximations, saddle point theorems, hemi-variationalinequalities, Nash equilibrium, etc. The vector equilibrium problem, containingmulti-objective optimization and vector variational inequalities, is an importantextension for the equilibrium problem. In addition, the equilibrium problem alsohas been extended more case, such as generalized equilibrium problems, system ofequilibrium problems, quasi-equilibrium problems and like-equilibrium problems.They have close ties with some non-linear analysis problems in optimization, con-trol theory, game theory, engineering and mechanics. And then they have wideapplication background.
This paper studies the solution existence for vector equilibrium problems,system of equilibrium problems and generalized equilibrium problems and searchessolving methods for them. The main points of this paper are as follows:
1. Based on Gerstewitz nonlinear scalarization function and the nonlin-ear scalarization function in variable control structure, the concept of nonlinearscalarization function for subsets of ordered topological vector space is presented.Some properties are investigated. The concept of upper semi-continuous-like ofset-valued mappings in topological spaces, which extended the concept of uppersemi-continuous of set-valued mappings, is proposed. The relationships betweenupper semi-continuous and upper semi-continuous-like is discussed.
2. The solution existence of the vector equilibrium problem in topologicalvector spaces is studied. An Iterative algorithm for the vector equilibrium prob- lem in European spaces, which is a vector version of Alfredo’s projection iterativealgorithm for the scalar equilibrium problem, is established. By using the non-linear scalarization method, the convergence of the algorithm is proved. As anextension of Tada’s algorithm for the scalar equilibrium problem, two algorithmsfor vector equilibrium problems in Hilbert spaces are proposed. Using the non-linear scalarization method, it is proved that one of the algorithms convergesstrongly and the other converges weakly.
3. As an extension for the well-known Ekland’s variational principle, the gen-eralized Ekland’s variational principle of set-valued mappings which includes thevector Ekland’s variational principle is gained. System of vector equilibrium prob-lems in metric spaces are studied. By using the generalized Ekland’s variationalprinciple of set-valued mappings, some new conclusions of solution existence forthe system of vector equilibrium problems are obtained. The conclusions are onlyrequired the functions in the system of vector equilibrium problems satisfying acertain continuity, without the domain satisfies any convexity, nor does it requirethe functions satisfies the monotonicity and generalized convexity conditions.
4. A hybrid iterative scheme for finding the common elements of the fixedpoints set for an infinite family of nonexpansive mappings and the set of solutionsfor a system of mixed equilibrium problems, which is an extension for Moudafi’sviscosity iterative method and explicit viscosity iterative scheme, is introduced.As corollaries, the algorithms for solving system of equilibrium problems andequilibrium problems are obtained.
5. The generalized vector equilibrium problems for set-valued mappings inmetric spaces are studied. By using the generalized Ekland’s variational princi-ple of set-valued mappings, some existence theorems are obtained. An iterativeprojection algorithm for solving the generalized vector equilibrium problem inEuropean spaces with variable control structure is presented. This algorithmis a generalization of Alfredo’s projection algorithm for the scalar equilibriumproblem.
6. As an extension for equilibrium problems, vector equilibrium problemsand generalized vector equilibrium problems in metric spaces, a kind of gener- alized equilibrium problem in a G-convex space is introduced. By means of thefixed-point theorems, some existence theorems with upper semi-continuous-likeset-valued mappings are obtained. As corollaries, some new existence results forequilibrium problems, vector equilibrium problems and generalized vector quasi-equilibrium problems are obtained.
本文主要研究向量均衡问题、均衡系统问题和广义均衡问题解的存在性,讨论其求解算法.具体如下：
1.基于Gerstewitz非线性标量化函数和变动控制结构下的非线性标量函数,引入了序拓扑向量空间中集合的非线性标量化函数概念,讨论了其性质.对拓扑空间上集值映射的连续性做了推广,引入了集值映射的类上半连续概念,讨论了它和集值映射上半连续的关系.对著名的Ekland变分原理进行了推广,得到了集值映射形式的广义Ekland变分原理,它包含了向量形式的Ekland变分原理.
2.研究了拓扑向量空间上向量均衡问题解的存在性,建立了欧氏空间上向量均衡问题的迭代算法,用非线性标量化方法证明了该算法的收敛性.该算法是对数量均衡问题的Alfredo投影迭代算法的向量形式推广.对Tada提出的Hilbert空间上数量均衡问题的算法进行了推广,得到了Hilbert空间上向量均衡问题的两个算法.用非线性标量化方法证明了其中一个算法强收敛,而另一个算法是弱收敛的.
3.研究度量空间上的向量均衡系统问题.利用集值映射型广义Ekland变分原理,证明了向量均衡系统新的解的存在性结论.该结论只要求向量均衡系统中函数的满足某种连续性,而无需均衡系统的有效域满足任何凸性,也不要求函数满足单调性及广义凸性条件.
4.对Moudafi的粘性迭代方法和显性粘性迭代方法进行了推广,构造了一个混合迭代方案,用于求解Hilbert空间上一个混合均衡系统问题和无限个非扩张映射的不动点问题的公共解,作为推论得到了Hilbert空间上均衡系统问题和均衡问题的求解算法.
5.研究了度量空间上的集值映射型广义向量均衡问题,用集值映射型广义Ekland变分原理建立了其解的存在性定理.构造了欧氏空间上具有变动控制结构的广义向量均衡问题的一个投影迭代算法,它是Alfredo数量均衡问题投影迭代算法的一个推广
6.在广义凸空间上定义了一类新的广义均衡问题,研究了其解的存在性,利用不动点定理证明了满足集值映射类上半连续性条件的解的存在性定理.作为该定理的推论,推导出了均衡问题、向量均衡问题和广义向量拟均衡问题解的存在性的一些新结果.
Equilibrium is a central concept to study some systems governing many real-life phenomena in fields ranging from economics and networks to mechanics. Andthe study of equilibrium in real-world applications have promoted the develop-ment of fixed point theory and optimization. The abstract variational inequality isa simple and natural formulation for those systems. Ky Fan’s minimax inequalityis another statement. And the equilibrium problem formulation is a generaliza-tion for them. The equilibrium problem formulation not only proposed a uni-fied model including optimization, fixed point theorems, variational inequalitiesand complementarity problems, but also contained other important mathemati-cal models, such as best approximations, saddle point theorems, hemi-variationalinequalities, Nash equilibrium, etc. The vector equilibrium problem, containingmulti-objective optimization and vector variational inequalities, is an importantextension for the equilibrium problem. In addition, the equilibrium problem alsohas been extended more case, such as generalized equilibrium problems, system ofequilibrium problems, quasi-equilibrium problems and like-equilibrium problems.They have close ties with some non-linear analysis problems in optimization, con-trol theory, game theory, engineering and mechanics. And then they have wideapplication background.
This paper studies the solution existence for vector equilibrium problems,system of equilibrium problems and generalized equilibrium problems and searchessolving methods for them. The main points of this paper are as follows:
1. Based on Gerstewitz nonlinear scalarization function and the nonlin-ear scalarization function in variable control structure, the concept of nonlinearscalarization function for subsets of ordered topological vector space is presented.Some properties are investigated. The concept of upper semi-continuous-like ofset-valued mappings in topological spaces, which extended the concept of uppersemi-continuous of set-valued mappings, is proposed. The relationships betweenupper semi-continuous and upper semi-continuous-like is discussed.
2. The solution existence of the vector equilibrium problem in topologicalvector spaces is studied. An Iterative algorithm for the vector equilibrium prob- lem in European spaces, which is a vector version of Alfredo’s projection iterativealgorithm for the scalar equilibrium problem, is established. By using the non-linear scalarization method, the convergence of the algorithm is proved. As anextension of Tada’s algorithm for the scalar equilibrium problem, two algorithmsfor vector equilibrium problems in Hilbert spaces are proposed. Using the non-linear scalarization method, it is proved that one of the algorithms convergesstrongly and the other converges weakly.
3. As an extension for the well-known Ekland’s variational principle, the gen-eralized Ekland’s variational principle of set-valued mappings which includes thevector Ekland’s variational principle is gained. System of vector equilibrium prob-lems in metric spaces are studied. By using the generalized Ekland’s variationalprinciple of set-valued mappings, some new conclusions of solution existence forthe system of vector equilibrium problems are obtained. The conclusions are onlyrequired the functions in the system of vector equilibrium problems satisfying acertain continuity, without the domain satisfies any convexity, nor does it requirethe functions satisfies the monotonicity and generalized convexity conditions.
4. A hybrid iterative scheme for finding the common elements of the fixedpoints set for an infinite family of nonexpansive mappings and the set of solutionsfor a system of mixed equilibrium problems, which is an extension for Moudafi’sviscosity iterative method and explicit viscosity iterative scheme, is introduced.As corollaries, the algorithms for solving system of equilibrium problems andequilibrium problems are obtained.
5. The generalized vector equilibrium problems for set-valued mappings inmetric spaces are studied. By using the generalized Ekland’s variational princi-ple of set-valued mappings, some existence theorems are obtained. An iterativeprojection algorithm for solving the generalized vector equilibrium problem inEuropean spaces with variable control structure is presented. This algorithmis a generalization of Alfredo’s projection algorithm for the scalar equilibriumproblem.
6. As an extension for equilibrium problems, vector equilibrium problemsand generalized vector equilibrium problems in metric spaces, a kind of gener- alized equilibrium problem in a G-convex space is introduced. By means of thefixed-point theorems, some existence theorems with upper semi-continuous-likeset-valued mappings are obtained. As corollaries, some new existence results forequilibrium problems, vector equilibrium problems and generalized vector quasi-equilibrium problems are obtained.
引文
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