In this work, we deal with Truncated Newton methods for solving large scale (possibly nonconvex) unconstrained optimization problems. In particular, we consider the use of a modified Bunch and Kaufman factorization for solving the Newton equation, at each (outer) iteration of the method. The Bunch and Kaufman factorization of a tridiagonal matrix is an effective and stable matrix decomposition, which is well exploited in the widely adopted SYMMBK (Bunch and Kaufman in Math Comput 31:163–179, 1977; Chandra in Conjugate gradient methods for partial differential equations, vol 129, 1978; Conn et al. in Trust-region methods. MPS-SIAM series on optimization, Society for Industrial Mathematics, Philadelphia, 2000; HSL, A collection of Fortran codes for large scale scientific computation, http://www.hsl.rl.ac.uk/; Marcia in Appl Numer Math 58:449–458, 2008) routine. It can be used to provide conjugate directions, both in the case of \(1\times 1\) and \(2\times 2\) pivoting steps. The main drawback is that the resulting solution of Newton’s equation might not be gradient–related, in the case the objective function is nonconvex. Here we first focus on some theoretical properties, in order to ensure that at each iteration of the Truncated Newton method, the search direction obtained by using an adapted Bunch and Kaufman factorization is gradient–related. This allows to perform a standard Armijo-type linesearch procedure, using a bounded descent direction. Furthermore, the results of an extended numerical experience using large scale CUTEst problems is reported, showing the reliability and the efficiency of the proposed approach, both on convex and nonconvex problems.

在这项工作中，我们处理了截断牛顿法，以解决大规模（可能是非凸）无约束优化问题。特别是，我们考虑在方法的每次（外部）迭代中使用改进的Bunch和Kaufman分解来求解牛顿方程。三对角矩阵的Bunch和Kaufman分解是一种有效且稳定的矩阵分解，在广泛采用的SYMMBK（Bunch和Kaufman在Math Comput 31：163-179，1977； Chandra在共轭梯度法中用于偏微分方程）中得到了很好的利用。 ，第129卷，1978年； Conn等人在“信赖区方法”中，关于优化的MPS-SIAM系列，工业数学协会，费城，2000年； HSL，用于大规模科学计算的Fortran代码集合，http：// www .hsl.rl.ac.uk /; Marcia in Appl Numer Math 58：449–458，2008年）。在以下情况下，都可以用来提供共轭方向\（1 \ times 1 \）和\（2 \ times 2 \）旋转步骤。主要缺点是，如果目标函数是非凸的，则牛顿方程的所得解可能与梯度无关。在这里，我们首先关注一些理论属性，以确保在Truncated Newton方法的每次迭代中，通过使用自适应Bunch和Kaufman分解获得的搜索方向与梯度相关。这允许使用有界下降方向执行标准Armijo型线搜索过程。此外，报告了使用大规模CUTEst问题的扩展数值经验的结果，显示了所提出方法在凸问题和非凸问题上的可靠性和有效性。