两类变分不等式系统解的存在性和解的迭代算法

发布时间：2018-01-07 09:28

本文关键词：两类变分不等式系统解的存在性和解的迭代算法　出处：《重庆师范大学》2015年硕士论文　论文类型：学位论文

【摘要】：变分不等式理论是应用数学中一个十分重要的研究领域,它在非线性最优化理论、微分方程、控制论、对策论、社会经济平衡理论等领域有着广泛的应用。变分不等式系统作为变分不等式的一种重要的推广,在近些年已经成为众多学者研究的热点问题之一。为此本文主要研究了定义在Hilbert空间中的两类特殊的变分不等式系统问题：一类定义在Hilbert空间中的新的广义非线性变分不等式系统问题和一类定义在Hilbert空间中的一致临近正则集上的带有四个非线性算子的广义非凸变分不等式系统问题。本文考查了这两类变分不等式系统问题的解的存在性和迭代算法。本文其余部分的布局如下：第一章,概述变分不等式理论研究的历史背景及研究现状,并简要介绍了本论文的一些研究工作。第二章,介绍了有关变分不等式问题,变分不等式问题解的存在性、唯一性问题,变分不等式问题的求解算法等基础理论。第三章,考虑一类定义在Hilbert空间中的新的广义非线性变分不等式系统问题,简记为：SGNLVIP。本文建立了SGNLVIP和不动点问题之间的等价性。证明了SGNLVIP解的存在性,并利用预解算子方法,对SGNLVIP提出一些新的显示平行迭代算法,在适当的假设条件下证明了这些算法的收敛性。第四章,考虑一类定义在Hilbert空间中的一致临近正则集上的带有四个非线性算子的广义非凸变分不等式系统问题,简记为：SGNCVIP。建立了SGNCVIP和不动点问题之间的等价性,并利用这种等价性提出了求解SGNCVIP的平行投影算法。在适当的条件下,我利用投影方法证明了SGNCVIP解的存在性问题和解的逼近问题。最后,给出了求解SGNCVIP另一种新的平行投影算法,该算法收敛至SGNCVIP的解集和两个Lipschitzian映像的交集,并且在适当的条件下考虑了该算法的收敛性。第五章,对本文的研究内容进行了简要的总结,并给出了后续的一些可以考虑的研究工作。
[Abstract]:Variational inequality theory is an important research field in Applied Mathematics, the theory of nonlinear optimization, differential equations, control theory, game theory, social economic equilibrium theory is widely used in many fields. The variational inequality as an important extension of the variational inequality in recent years has become a hot topic the problem for many scholars. This paper mainly studies the definition in the Hilbert space of the two kinds of special system of variational inequalities problem: a class of generalized nonlinear Hilbert in space in the new system of variational inequality problems and a class definition in the Hilbert space on the regular set near consistent with four nonlinear the operator of generalized non convex variational inequality problems. This paper examines the system of these two types of systems of variational inequalities problem and iterative algorithm. The rest of the cloth The bureau is as follows: the first chapter, an overview of the variational inequality theory historical background and current situation of research, and briefly describes some of the research work in this thesis. The second chapter introduces the variational inequality problem, solution existence of variational inequalities, the only problem based algorithms for solving variational inequalities theory. In the third chapter, we consider a class of generalized nonlinear definition in the Hilbert space of the new system of variational inequalities problems, such as: SGNLVIP. this paper establishes the equivalence between SGNLVIP and fixed point problems. SGNLVIP solution is proved the existence of solutions, and the use of pre operator method, put forward some new display parallel iterative algorithm for SGNLVIP, the convergence of these algorithms is proved under appropriate conditions. In the fourth chapter, a class definition in the Hilbert space in the same near regular set with four wide nonlinear operator The meaning of non convex variational inequality system, abbreviated as SGNCVIP. to establish the equivalence between the SGNCVIP and the fixed point problem, and using this equivalence proposed parallel projection algorithm for solving SGNCVIP. Under appropriate conditions, I use the projection method proved the existence problem of approximation solution and SGNCVIP. Finally, SGNCVIP presented a new parallel projection algorithm, the algorithm converges to the intersection of SGNCVIP solution and two Lipschitzian images, and consider the convergence of the algorithm under suitable conditions. The fifth chapter, the research contents of this paper are summarized, and some can be considered the following research work.

【学位授予单位】：重庆师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O178

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