非线性最小二乘问题收敛性的证明  

A Proof on Convergence Properies of Nonlinear Least Squares Problem

作　　者：陈宏彩[1] 张友兰[1] 

机构地区：[1]河北省科学院应用数学研究所,河北石家庄050081

出　　处：《数学的实践与认识》2010年第23期149-154,共6页Mathematics in Practice and Theory

摘　　要：主要讨论了无约束最优化中非线性最小二乘问题的收敛性.侧重于收敛的速率和整体、局部分析.改变了Gauss—Newton方法收敛性定理的条件,分两种情况证明了:(1)目标函数的海赛矩阵正定(函数严格凸)时为强整体二阶收敛;(2)目标函数不保证严格凸性,但海赛矩阵的逆存在时为局部收敛,敛速仍为二阶,同时给出了J(X)^(-1)和Q(X)^(-1)之间存在、有界性的等价条件.This paper deals mainly with convergence properties of nonlinear least squares problem in unconstrained optimization.lay particular emphasis on analysis of global and local convergence and rate of convergence.Hypotheses on theorem of Gauss-Newton convergence properties have been changed.The proof is divided into two parts：（1）When Hessian matrix is positive definite{X^K}has strong global convergence of superlinear,order of convergence is at least 2.（2） When Hessian matrix is nonsingnlar{X^K}has local convergence of superlinear,order of convergence is at least 2.At the same time,the conditions of equivalence aboutQ（X）^（-1） and J（X）^（-1） having existential and bounded properties have been given.

关 键 词：收敛性 高斯—牛顿算法 海赛矩阵 

分 类 号：O241.5[理学—计算数学]