NEWTON-TYPE MINIMIZATION VIA THE LANCZOS METHOD
NEWTON-TYPE MINIMIZATION VIA THE LANCZOS METHOD
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DOI：
10.1137/0721052
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发表时间：
1984-01-01
影响因子：
2.9
通讯作者：
NASH, SG
中科院分区：
文献类型：
--
作者：
NASH, SG
本文讨论了线性共轭梯度法（由Lanczos方法发展而来）在求解大规模无约束极小化问题中的应用。在牛顿型方法的每次迭代中，搜索方向被定义为二次子问题的解。当变量的数目非常大时，这个子问题可以使用Hestenes和Stiefel的线性共轭梯度法来解决。我们展示了如何利用线性共轭梯度法的等价Lanczos特征来定义一个修改后的牛顿法，该方法可以应用于不一定在感兴趣的区域的所有点都具有正定Hessian矩阵的问题。这个推导也使得在一个静止点计算负曲率方向成为可能。截断牛顿法的思想是只执行有限次数的二次子问题的迭代。这有效地给出了在最速下降方向和牛顿方向之间插值的搜索方向。我们描述了一个预处理的线性共轭梯度法，定义了一个搜索方向之间的插值定义的方向由非线性共轭梯度型算法和修改后的牛顿方向。
This paper discusses the use of the linear conjugate-gradient method (developed via the Lanczos method) in the solution of large-scale unconstrained minimization problems. At each iteration of a Newton-type method, the direction of search is defined as the solution of a quadratic subproblem. When the number of variables is very large, this subproblem may be solved using the linear conjugate-gradient method of Hestenes and Stiefel. We show how the equivalent Lanczos characterization of the linear conjugate-gradient method may be exploited to define a modified Newton method which can be applied to problems that do not necessarily have positive-definite Hessian matrices at all points of the region of interest. This derivation also makes it possible to compute a negative-curvature direction at a stationary point.The idea of a truncated Newton method is to perform only a limited number of iterations of the quadratic subproblem. This effectively gives a search direction that interpolates between the steepest-descent direction and the Newton direction. We describe a preconditioned linear conjugate-gradient method that defines a search direction which interpolates between the direction defined by a nonlinear conjugate-gradient-type algorithm and a modified Newton direction.