Newton-type minimization via the Lanczos method 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. It is shown 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. This derivation also makes it possible to compute a negative-curvature direction at a stationary point. The above mentioned modified Lanczos algorithm requires up to n iterations to compute the search direction, where n denotes the number of variables of the problem. The idea of a truncated Newton method is to terminate the iterations earlier. A preconditioned truncated Newton method is described that defines a search direction which interpolates between the direction defined by a nonlinear conjugate-gradient-type method and a modified Newton direction. Numerical results are given which show the promising performance of truncated Newton methods. (Source:

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  11. Brás, C. P.; Martínez, J. M.; Raydan, M.: Large-scale unconstrained optimization using separable cubic modeling and matrix-free subspace minimization (2020)
  12. Caliciotti, Andrea; Fasano, Giovanni; Potra, Florian; Roma, Massimo: Issues on the use of a modified bunch and Kaufman decomposition for large scale Newton’s equation (2020)
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  14. De Leone, Renato; Fasano, Giovanni; Roma, Massimo; Sergeyev, Yaroslav D.: Iterative grossone-based computation of negative curvature directions in large-scale optimization (2020)
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  20. Busseti, Enzo; Moursi, Walaa M.; Boyd, Stephen: Solution refinement at regular points of conic problems (2019)

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