Issues on the use of a modified Bunch and Kaufman decomposition for large scale Newton’s equation

Abstract

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 1×1 and 2×2 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.