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1、 变分包含问题论文：求解变分包含问题的迫近点算法和近似束方法【中文摘要】本文的主要内容可概括如下:第一章中,针对Banach空间的一类非线性变分包含问题,将文献中Hilbert空间的A-极大单调映射进行一般推广,提出了Banach空间的( A,)-极大增生算子的概念.通过研究( A,)-极大增生算子的性质,改进了与A-极大单调映射相关的预解算子技巧,将其推广为与( A,)-极大增生算子相关的新型预解算子.在本章的最后讨论了新型预解算子的有关性质.在第一章的基础上,第二章主要考虑非线性包含问题0M(x)的解的存在性和唯一性. 2007年, Ram U. Verma结合( A,)-极大单调算子,提
2、出了这类变分包含问题的混合迫近点算法框架.本论文在此基础上,应用( A,)-极大增生算子,对文献中的混合迫近点算法一般框架进行了推广和改进,提出一种新型迭代算法.同时,应用预解算子的相关结论对求解变分包含问题的混合迫近点算法进行了收敛性分析,所得的结论将非线性变分包含问题相关结果推广为涉及( A,)-极大增生算子的非线性变分包含问题的结果.第三章中,为了解决广义变分不等式的求解问题,我们考虑附属问题原则的一种推广,将非光滑优化中的束方法思想与解变分不等式的辅助问题方法相结合,提出了一种解广义变分不等式的近似束-型辅助问题方法.所讨论的问题是求解两个定义在实的Hilbert空间上的算子之和的零点
3、:第一个算子是一个单调的单值算子;第二个是一个下半连续的正常凸函数的次微分.算法构造中,对辅助函数的要求减弱了,不再要求强凸,只要凸就可以了.最后我们证明了在一定条件下算法的弱收敛性.【英文摘要】The contents of this paper can be divided into three parts.In the first part, for a class of nonlinear variational inclusion problems in Banach spaces, we introduce the notion of ( A,)-maximal accretive
4、, which is a generalization of the A -maximal monotonicity mapping in Hilbert spaces. We extend the resolvent operator technique by studying the properties of ( A,)-maximal accretive operators and generalize the concept of resolvent operators associated with the A -maximal monotonicity mappings to t
5、he one associated with ( A,)-maximal accretive operators. By using the new resolvent operator technique, we prove the existence and uniqueness of the solution to a class of nonlinear vatiational inclusion problems. At the end of this part, we discuss the properties of this new resolvent operator.Nex
6、t, in the second part, a general framework for a hybrid proximal point algorithm using the notion of ( A,)-maximal accretive is proposed. Convergence analysis for this algorithm in the context of solving a class of nonlinear varitional inclusion problems is also explored. The obtained results genera
7、lize the conclusions of nonlinear variational inclusions in 1 to the one associated with ( A,)-maximal accretive operators.In the final part, for solving a class of generalized variational inequalities problems, auxiliary principle is generalized. By combining the bundle methods with auxiliary princ
8、iples, we propose an approximate bundle-type auxiliary principle method for the considered problem which is to find the zero point of the sum of two operators, one is monotonic single-valued, and the other is the subdifferential of a low semi continuous, proper, convex function. During the construct
9、ion of the algorithm, the condition imposed on the auxiliary function is weakened, not necessarily strongly convex, convex is enough. Finally, we prove the weak convergence of the proposed algorithm.【关键词】变分包含问题 混合迫近点算法 ( A,)-极大增生 束方法【英文关键词】variational inclusion problem hybrid proximal point algorith
10、m ( A,)-maximal accretive Bundle methods【目录】求解变分包含问题的迫近点算法和近似束方法摘要4-5Abstract5引言7-91 预备知识9-121.1 (A, ) -极大增生9-101.2 预解算子10-122 混合迫近点算法12-172.1 理论基础12-142.2 算法构造14-173 束方法在广义变分不等式中的应用17-313.1 束方法的由来与发展17-193.2 问题的产生19-203.3 求解（GVIP）的广义近似算法20-223.4 算法构造22-233.5 收敛性分析23-31结论31-32参考文献32-36攻读硕士学位期间发表学术论文情况36-37致谢37
