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. Zhou, W.; Akrotirianakis, I. G.; Yektamaram, S.; Griffin, J. D.: A matrix-free line-search algorithm for nonconvex optimization (2019)
  2. Campos, Juan S.; Parpas, Panos: A multigrid approach to SDP relaxations of sparse polynomial optimization problems (2018)
  3. Livieris, Ioannis E.; Tampakas, Vassilis; Pintelas, Panagiotis: A descent hybrid conjugate gradient method based on the memoryless BFGS update (2018)
  4. Salim, M. S.; Ahmed, A. I.: A family of quasi-Newton methods for unconstrained optimization problems (2018)
  5. Fasano, Giovanni; Pesenti, Raffaele: Conjugate direction methods and polarity for quadratic hypersurfaces (2017)
  6. Grote, Marcus J.; Kray, Marie; Nahum, Uri: Adaptive eigenspace method for inverse scattering problems in the frequency domain (2017)
  7. 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)
  8. Métivier, L.; Brossier, R.; Operto, S.; Virieux, J.: Full waveform inversion and the truncated Newton method (2017)
  9. Di, Zichao (Wendy); Leyffer, Sven; Wild, Stefan M.: Optimization-based approach for joint X-ray fluorescence and transmission tomographic inversion (2016)
  10. Gratton, Serge; Mercier, Sylvain; Tardieu, Nicolas; Vasseur, Xavier: Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics. (2016)
  11. Letham, Benjamin; Letham, Portia A.; Rudin, Cynthia; Browne, Edward P.: Prediction uncertainty and optimal experimental design for learning dynamical systems (2016)
  12. 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)
  13. Xu, Wei; Zheng, Ning; Hayami, Ken: Jacobian-free implicit inner-iteration preconditioner for nonlinear least squares problems (2016)
  14. Fasano, Giovanni: A framework of conjugate direction methods for symmetric linear systems in optimization (2015)
  15. De Simone, V.; di Serafino, D.: A matrix-free approach to build band preconditioners for large-scale bound-constrained optimization (2014)
  16. Li, Xiangrong; Wang, Xiaoliang; Duan, Xiabin: A limited memory BFGS method for solving large-scale symmetric nonlinear equations (2014)
  17. Lukšan, Ladislav; Vlček, Jan: Efficient tridiagonal preconditioner for the matrix-free truncated Newton method (2014)
  18. O’Malley, D.; Vesselinov, V. V.; Cushman, J. H.: A method for identifying diffusive trajectories with stochastic models (2014)
  19. Fasano, Giovanni; Roma, Massimo: Preconditioning Newton-Krylov methods in nonconvex large scale optimization (2013)
  20. Kojima, Masakazu; Yamashita, Makoto: Enclosing ellipsoids and elliptic cylinders of semialgebraic sets and their application to error bounds in polynomial optimization (2013)

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