两类变分不等式的投影型方法研究

发布时间：2018-06-23 22:20

本文选题：混合变分不等式 + 对偶变分不等式　；参考：《广西民族大学》2016年硕士论文

【摘要】：投影型方法是求解变分不等式问题最重要的方法之一,本文主要研究求解两类变分不等式的投影型方法。全文分为四章,我们概括如下:第一章,介绍了本文的研究背景和基本定义及基本概念。第二章,在欧氏空间上构造出了一个求解混合变分不等式的投影型算法.该算法中的Armijo线性搜索与何诣然在变分不等式中所提出的相类似.在适当的条件下,该算法生成的序列将全局收敛到混合变分不等式问题的一个解.此外,在一个确定的误差界成立的情况下,我们分析了该算法生成的迭代序列的收敛率.第三章,在欧氏空间空间上构造出了一个求解广义变分不等式的投影型算法.在假设对偶变分不等式有解的情况下,我们得出该算法是良定的且由该算法生成的序列收敛到广义变分不等式的一个解.这个假设条件比其他作者使用算子的伪单调要弱很多.我们将提出一个例子来支持这个结果.对比于Li和He及Fang和He近段时间在该方面的研究结果,我们剔除了条件(A3).第四章,在Banach空间上构造了求解混合变分不等式的一个投影型算法.混合变分不等式解的存在性可由该算法生成的序列来鉴别.如果混合变分不等式的解集非空,那么由该算法生成的序列强收敛到(Bregman度量意义下)离初始点最近的一个解;否则,该序列是发散的.
[Abstract]:Projection method is one of the most important methods for solving variational inequality problems. The thesis is divided into four chapters. The first chapter introduces the research background, basic definitions and basic concepts. In chapter 2, we construct a projection algorithm for solving mixed variational inequalities in Euclidean space. The Armijo linear search in this algorithm is similar to that proposed by he Ji-ran in variational inequalities. Under suitable conditions, the sequence generated by the algorithm converges globally to a solution of the mixed variational inequality problem. In addition, if a definite error bound is established, the convergence rate of the iterative sequence generated by the algorithm is analyzed. In chapter 3, we construct a projection algorithm for solving generalized variational inequalities in Euclidean space. Under the assumption that there are solutions to dual variational inequalities, we obtain that the algorithm is well-defined and the sequence generated by the algorithm converges to a solution of generalized variational inequalities. This assumption is much weaker than the pseudo-monotone of the operator used by other authors. We will provide an example to support this result. Compared with the results of Li and he, Fang and he in the recent period, we exclude the condition (A3). In chapter 4, we construct a projection algorithm for solving mixed variational inequalities in Banach space. The existence of solutions to mixed variational inequalities can be identified by the sequence generated by the algorithm. If the solution set of mixed variational inequalities is not empty, the sequence generated by the algorithm converges strongly to the solution nearest to the initial point (in the sense of Bregman metric); otherwise, the sequence is divergent.
【学位授予单位】：广西民族大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O178;O176

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