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. Andrei, Neculai: A diagonal quasi-Newton updating method for unconstrained optimization (2019)
  2. Andrei, Neculai: A new diagonal quasi-Newton updating method with scaled forward finite differences directional derivative for unconstrained optimization (2019)
  3. Austin, Anthony P.; Di, Zichao; Leyffer, Sven; Wild, Stefan M.: Simultaneous sensing error recovery and tomographic inversion using an optimization-based approach (2019)
  4. 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)
  5. Zhou, W.; Akrotirianakis, I. G.; Yektamaram, S.; Griffin, J. D.: A matrix-free line-search algorithm for nonconvex optimization (2019)
  6. 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)
  7. Campos, Juan S.; Parpas, Panos: A multigrid approach to SDP relaxations of sparse polynomial optimization problems (2018)
  8. Livieris, Ioannis E.; Tampakas, Vassilis; Pintelas, Panagiotis: A descent hybrid conjugate gradient method based on the memoryless BFGS update (2018)
  9. Salim, M. S.; Ahmed, A. I.: A family of quasi-Newton methods for unconstrained optimization problems (2018)
  10. Fasano, Giovanni; Pesenti, Raffaele: Conjugate direction methods and polarity for quadratic hypersurfaces (2017)
  11. Grote, Marcus J.; Kray, Marie; Nahum, Uri: Adaptive eigenspace method for inverse scattering problems in the frequency domain (2017)
  12. 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)
  13. Métivier, L.; Brossier, R.; Operto, S.; Virieux, J.: Full waveform inversion and the truncated Newton method (2017)
  14. Di, Zichao (Wendy); Leyffer, Sven; Wild, Stefan M.: Optimization-based approach for joint X-ray fluorescence and transmission tomographic inversion (2016)
  15. Fasano, Giovanni; Roma, Massimo: A novel class of approximate inverse preconditioners for large positive definite linear systems in optimization (2016)
  16. Gratton, Serge; Mercier, Sylvain; Tardieu, Nicolas; Vasseur, Xavier: Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics. (2016)
  17. Letham, Benjamin; Letham, Portia A.; Rudin, Cynthia; Browne, Edward P.: Prediction uncertainty and optimal experimental design for learning dynamical systems (2016)
  18. Nita, C.; Vandewalle, S.; Meyers, J.: On the efficiency of gradient based optimization algorithms for DNS-based optimal control in a turbulent channel flow (2016)
  19. Xu, Wei; Zheng, Ning; Hayami, Ken: Jacobian-free implicit inner-iteration preconditioner for nonlinear least squares problems (2016)
  20. Fasano, Giovanni: A framework of conjugate direction methods for symmetric linear systems in optimization (2015)

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