Gauss-Newton BFGS方法所产生的迭代矩阵序列的收敛性
The Convergence of Iterate Matrices Sequences Generated by Gauss-Newton BFGS Methods
摘要: 收敛速度的快慢是决定一个算法好坏的重要因素。在拟牛顿算法中，算法的收敛性在某种程度上等价于Dennis-Moré条件，但这并不意味着算法所产生的迭代矩阵就会收敛到Hessian矩阵。本文证明了由求解对称非线性方程组的Gauss-Newton BFGS方法所产生的迭代矩阵序列的收敛性，并通过数值实验对结论进行验证。
Abstract:
The speed of convergence is an important factor that determines the quality of an algorithm. In the quasi-Newton algorithm, the convergence of the algorithm is equivalent to the Dennis-Moré condi-tion to some extent, but this does not mean that the iterative matrix generated by the algorithm will converge to the Hessian matrix. This paper proves the convergence of the iterative matrix se-quence generated by the Gauss-Newton BFGS method for solving symmetric nonlinear equations, and validates the conclusion by numerical experiments.
参考文献
|
[1]
|
Li, D. and Fukushima, M. (1999) A Globally and Superlinearly Convergent Gauss—Newton-Based BFGS Method for Symmetric Nonlinear Equations. SIAM Journal on numerical Analysis, 37, 152-172. [Google Scholar] [CrossRef]
|
|
[2]
|
Dennis, J.E. and Moré, J.J. (1974) A Characterization of Super-linear Convergence and Its Application to Quasi-Newton Methods. Mathematics of Computation, 28, 549-560. [Google Scholar] [CrossRef]
|
|
[3]
|
Ren-Pu, G. and Powell, M.J. (1983) The Convergence of Variable Metric Matrices in Unconstrained Optimization. Mathematical Programming, 27, 123-143. [Google Scholar] [CrossRef]
|
|
[4]
|
Stoer, J. (1984) The Convergence of Matrices Generated by Rank-2 Methods from the Restricted β-Class of Broyden. Numerische Mathematik, 44, 37-52. [Google Scholar] [CrossRef]
|
|
[5]
|
Fletcher, R. (1970) A New Approach to Variable Metric Algorithms. The Computer Journal, 13, 317-322. [Google Scholar] [CrossRef]
|
|
[6]
|
Powell, M.J. (1978) The Convergence of Variable Metric Methods for Nonlinearly Constrained Optimization Calculations. In: Nonlinear Programming 3, Academic Press, 27-63. [Google Scholar] [CrossRef]
|