文档介绍：致 谢
本文是在导师高雷阜教授的指导下完成的。跟随导师学****的两年半时间里,不但让我学到了相关的专业知识,同时也让我体会到了终身受益的科研方法和态度。在读期间,高老师悉心的关怀、渊博的知识和严谨的治学态度一直鼓励和感染着我,并让我受益良多。在这里,特别向高老师致以真挚的感谢和崇高的敬意。
再次,还要感谢每一位向我传授知识的老师。他们无私奉献的精神和严谨治学的态度让我在学****中的各方面收获很多。
最后要感谢师兄、师姐、师弟、师妹们,他们对我的关心和帮助,让我度过了美好的时光。
摘 要
文中主要对半定规划的若干算法进行了研究,其中包括预测校正算法和收缩映射算法。
首先介绍了半定规划的基础知识,主要包括标准的半定规划及其对偶理论,半定规划的两个主要算法: 内点法和谱丛算法,该理论在二次锥优化、二次规划中的应用等。
其次,从已有的预测校正算法对一类半定规划的子问题提出了针对该问题的相应的预测校正算法。并且通过数值试验,验证了提出的算法的可行性。
最后,结合半定规划与一类线性变分不等式之间的等价性,将半定规划转化为线性变分不等式。利用该线性变分不等式的一类投影映射算法间接求解半定规划,并在映射算法的基础上提出了一种基于电路的神经网络模型。而且,从理论上证明了该映射算法的稳定性和渐进稳定性。
关键词:半定规划;内点法;预测校正算法;变分不等式;投影映射
Abstract
Several algorithms for semidefinite programming are studied, which include the contraction mapping algorithm and predictor-corrector algorithm.
Firstly, the basics knowledge of SDP is introduced. These include the standard semidefinite programming, semidefinite programming duality theory, two main algorithms for semidefinite programming: interior point method and the spectral bundle algorithm, and then simply give the applications in the second optimization for the cone and quadratic programming.
Then, a predictor-corrector algorithm for a sub-class of semidefinite programming problem is proposed with the exsited theroy in the path following algorithm, and through numerical experiments the proposed algorithm is tested.
At last, with the help of the equivalent between semidefinite programming and a subclass of variational inequality and the equivalent between this variational inequality and projection mapping algorithm, projection mapping algorithm is used for solving semidefinite programming. Thus a circuit is proposed basing on the mapping algorithm, at the same time, the stability of the mapping algorithm and evolutionary convergence are proved theoretically too.
Key Words : Semidefinite Programming ; Interior Algorithm ; Predictor-Corrector
Algorithm;Variational Inequality;Projection method.
目 录
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