Dogleg路径信赖域方法

本文提出一类新的解无约束最优化问题的信整域方法。这类方法是通过对一般对称矩阵的Bunch-Parlett分解来产生搜索路径。它们既可以解目标函数是二次可微的也可以解目标函数是非二次可微的最优化问题，并且在由算法得到点列的任意聚点上，二次连续可微的目标函数的Hesse阵都是正定或半正定的。我们证明在一些较弱的条件下，算法是整体收敛的；对一致凸函数，是二次收敛的。一些数值结果表明这种新的方法是非常有效的。

A Class of Dogleg Trust Region Methods

In this paper, we propose a new class of trust region methods for nonlinear optimization problems. Our interest and motivation are to construct such a class of trust region methods which can be used for both twice and non-twice differentiable functions. We also want that the Hessian matrices of the objective function, if they exist, are positive definite or positive semidefinite at all accumulation points of {xk} obtained by the methods. We find an approximate solution δ* of the quadratic subproblem by piecewise linear paths called dogleg paths and obtained by employing Bunch-Parlett factorization for general symmetric matrices. We prove that these methods are convergent for continuous differentiable functions and quadratic for uniformly convex objective functions.