基于新拟牛顿方程改进的一类BFGS算法及其收敛性分析

发布时间：2019-01-14 10:11

【摘要】：以最小的付出获得最大的收益,这是研究任何事或者对某些事做决策时所追求的目标,而数学理论的研究往往都是将问题在合理的假设下建立相应的数学模型,将之转化为无约束最优化问题的求解。经过几十年的理论发展研究,解决这类问题最有效的方法就是拟牛顿法中的BFGS算法,本文在众多学者研究成果的基础上对该算法进行了改进,并取得较好的收敛速度与数值效果,具体研究内容如下:首先,在Biggs和Yuan所提kB校正公式的基础上,引入参数??[0,1]将两者相结合,提出一个改进的kB校正公式,当参数取两端点时退化为两位学者所提公式,并根据文献思路对本文所提校正公式做进一步必要说明以确保合理性,并根据改进公式给出改进的BFGS算法(MBFGS)。其次,将本文所提的kB校正公式与新拟牛顿方程相结合,提出一个基于新拟牛顿方程改进的BFGS算法(RMBFGS),并给出算法的收敛性证明,包括全局收敛性和局部超线性收敛性,同时进行数值实验证明算法要优于标准BFGS算法和同等改进的BFGS算法。最后,考虑到数据维度有不断增大的趋势,本文参照标准L-BFGS算法的思路,将MBFGS算法做进一步扩展,推导出适用于求解大规模无约束最优化问题的改进的L-BFGS算法(L-RMBFGS)。
[Abstract]:The goal of studying anything or making decisions about something is to get the most out of it with the least effort, and the study of mathematical theory is often to build a mathematical model of the problem under reasonable assumptions. It is transformed into an unconstrained optimization problem. After decades of theoretical development and research, the most effective method to solve this kind of problems is the BFGS algorithm in the quasi-Newton method. This paper improves the algorithm on the basis of many scholars' research results. A better convergence rate and numerical effect are obtained. The specific research contents are as follows: firstly, based on the kB correction formula proposed by Biggs and Yuan, an improved kB correction formula is proposed by combining the two parameters. When the parameters are taken at both ends, the formula is reduced to that proposed by two scholars, and the correction formula proposed in this paper is further explained according to the literature ideas to ensure the rationality, and the improved BFGS algorithm (MBFGS). Is given according to the improved formula. Secondly, combining the kB correction formula proposed in this paper with the new quasi-Newton equation, an improved BFGS algorithm (RMBFGS), based on the new quasi-Newton equation is proposed and the convergence proof of the algorithm is given, including global convergence and local superlinear convergence. At the same time, numerical experiments show that the algorithm is superior to the standard BFGS algorithm and the same improved BFGS algorithm. Finally, considering the increasing trend of data dimension, this paper makes further expansion of MBFGS algorithm according to the idea of standard L-BFGS algorithm. An improved L-BFGS algorithm (L-RMBFGS) for solving large scale unconstrained optimization problems is derived.
【学位授予单位】：西安建筑科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O224

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