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集值广义向量变分不等式问题的研究
On the Generalized Vector Variational Inequalities with Set-valued Mapping Problom
【作者】 于姝；
【导师】 夏尊铨；
【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2011， 硕士
【摘要】 集值映射广义向量变分不等值(GVVTI)是变分不等式的重要推广形式,是研究多目标规划、均衡问题以及其他数学和工程领域中等问题的重要理论基础和工具,对这一问题的研究涉及到集值分析、凸分析、线性与非线性分析、非光滑分析、泛函分析等,有重要的学术价值。本文主要从理论上研究Hausdorff拓扑向量空间上一类集值映射广义向量变分不等式解的存在性问题,通过添加下半连续、C-单调等条件,运用经典的KKM-Fan定理对其等价问题的解的存在性进行证明,这一结论也统一和推广了许多已有的向量变分不等式解的存在性问题的研究。在求解变分不等式时,我们常把一个变分不等式问题转化为一个最优化问题,通过解决最优化问题从而达到求解变分不等式的目的,而Gap函数在这种转化过程中起到了至关重要的作用。Gap函数最初应用在优化问题上并得到了一系列较好的应用,但是把Gap函数运用到变分不等式问题上还是近几年发展起来的一种趋势。所以本文就所提出的Hausdorff拓扑向量空间集值广义向量变分不等式模型定义了Gap函数,并给出了此类集值广义向量变分不等式解存在的充分必要条件。
【Abstract】 Generalize vector variational inequalities with set-valued mapping are very important generation of variational inequalities. They become significant foundation and tool for studying multiobjective program,equilibrium, as well as many problems in the fields of mathematics and engineering. Because generalized vector variational inequlitise touch upon many mathematical branches, such as applications in set-valued analysis,convex analysis, linear and nonlinear analysis,nonsmooth analysis, function analysis,the research for them is of much academic value. This dissertation is devoted to study the existence of solution of generalized vector variational inequalities with set-valued in Haudorff topological vector spaces by KKM-Fan theorem,through giving conditions such as lower semicontinuity and C-monotonicty. This result is also the unity and extension of a number of well-known existence theorem of solution of vector variational inequalities.When solving variational inequalities, we often transform a variational inequality problem into an optimization problem. Then we can solve the variational inequality by solving the optimization problem. Gap function plays a crucial role in this kind of transformation. Gap function is always used in the optimization problem on the initial application and get a series of good application, but the Gap function applied to the variational inequality problem is developed in recent years as a trend. So this paper will define a kind of Gap function for the set-valued generalized vector variational inequality in the Hausdorff topological vector space and will established necessary and sufficient conditions for the existence of a solution for the GVVI.