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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  1. Al-Baali, Mehiddin; Caliciotti, Andrea; Fasano, Giovanni; Roma, Massimo: A class of approximate inverse preconditioners based on Krylov-subspace methods for large-scale nonconvex optimization (2020)
  2. Andrei, Neculai: Diagonal approximation of the Hessian by finite differences for unconstrained optimization (2020)
  3. Brás, C. P.; Martínez, J. M.; Raydan, M.: Large-scale unconstrained optimization using separable cubic modeling and matrix-free subspace minimization (2020)
  4. Fung, Samy Wu; Di, Zichao: Multigrid optimization for large-scale ptychographic phase retrieval (2020)
  5. Andrei, Neculai: A diagonal quasi-Newton updating method for unconstrained optimization (2019)
  6. Andrei, Neculai: A new diagonal quasi-Newton updating method with scaled forward finite differences directional derivative for unconstrained optimization (2019)
  7. Austin, Anthony P.; Di, Zichao; Leyffer, Sven; Wild, Stefan M.: Simultaneous sensing error recovery and tomographic inversion using an optimization-based approach (2019)
  8. Busseti, Enzo; Moursi, Walaa M.; Boyd, Stephen: Solution refinement at regular points of conic problems (2019)
  9. Józsa, Tamas I.; Balaras, E.; Kashtalyan, M.; Borthwick, A. G. L.; Viola, I. M.: Active and passive in-plane wall fluctuations in turbulent channel flows (2019)
  10. Xu, Min; Zhou, Bojian; He, Jie: Improving truncated Newton method for the logit-based stochastic user equilibrium problem (2019)
  11. Zhou, W.; Akrotirianakis, I. G.; Yektamaram, S.; Griffin, J. D.: A matrix-free line-search algorithm for nonconvex optimization (2019)
  12. Caliciotti, Andrea; Fasano, Giovanni; Nash, Stephen G.; Roma, Massimo: An adaptive truncation criterion, for linesearch-based truncated Newton methods in large scale nonconvex optimization (2018)
  13. Campos, Juan S.; Parpas, Panos: A multigrid approach to SDP relaxations of sparse polynomial optimization problems (2018)
  14. Livieris, Ioannis E.; Tampakas, Vassilis; Pintelas, Panagiotis: A descent hybrid conjugate gradient method based on the memoryless BFGS update (2018)
  15. Salim, M. S.; Ahmed, A. I.: A family of quasi-Newton methods for unconstrained optimization problems (2018)
  16. Fasano, Giovanni; Pesenti, Raffaele: Conjugate direction methods and polarity for quadratic hypersurfaces (2017)
  17. Grote, Marcus J.; Kray, Marie; Nahum, Uri: Adaptive eigenspace method for inverse scattering problems in the frequency domain (2017)
  18. Mercier, Sylvain; Gratton, Serge; Tardieu, Nicolas; Vasseur, Xavier: A new preconditioner update strategy for the solution of sequences of linear systems in structural mechanics: application to saddle point problems in elasticity (2017)
  19. Métivier, L.; Brossier, R.; Operto, S.; Virieux, J.: Full waveform inversion and the truncated Newton method (2017)
  20. Di, Zichao (Wendy); Leyffer, Sven; Wild, Stefan M.: Optimization-based approach for joint X-ray fluorescence and transmission tomographic inversion (2016)

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