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. Di, Zichao (Wendy); Leyffer, Sven; Wild, Stefan M.: Optimization-based approach for joint X-ray fluorescence and transmission tomographic inversion (2016)
  2. Fasano, Giovanni: A framework of conjugate direction methods for symmetric linear systems in optimization (2015)
  3. De Simone, V.; di Serafino, D.: A matrix-free approach to build band preconditioners for large-scale bound-constrained optimization (2014)
  4. Lukšan, Ladislav; Vlček, Jan: Efficient tridiagonal preconditioner for the matrix-free truncated Newton method (2014)
  5. O’Malley, D.; Vesselinov, V.V.; Cushman, J.H.: A method for identifying diffusive trajectories with stochastic models (2014)
  6. Fasano, Giovanni; Roma, Massimo: Preconditioning Newton-Krylov methods in nonconvex large scale optimization (2013)
  7. Kojima, Masakazu; Yamashita, Makoto: Enclosing ellipsoids and elliptic cylinders of semialgebraic sets and their application to error bounds in polynomial optimization (2013)
  8. Métivier, L.; Brossier, R.; Virieux, J.; Operto, S.: Full waveform inversion and the truncated Newton method (2013)
  9. Armand, Paul; Benoist, Joël; Dussault, Jean-Pierre: Local path-following property of inexact interior methods in nonlinear programming (2012)
  10. Chouzenoux, E.; Moussaoui, S.; Idier, J.: Majorize-minimize linesearch for inversion methods involving barrier function optimization (2012)
  11. Kiiveri, Harri; de Hoog, Frank: Fitting very large sparse Gaussian graphical models (2012)
  12. Papadimitriou, D.I.; Giannakoglou, K.C.: Aerodynamic design using the truncated Newton algorithm and the continuous adjoint approach (2012)
  13. Byrd, Richard H.; Chin, Gillian M.; Neveitt, Will; Nocedal, Jorge: On the use of stochastic Hessian information in optimization methods for machine learning (2011)
  14. De Hoog, F.R.; Anderssen, R.S.; Lukas, M.A.: Differentiation of matrix functionals using triangular factorization (2011)
  15. Malmedy, Vincent; Toint, Philippe L.: Approximating Hessians in unconstrained optimization arising from discretized problems (2011)
  16. Yuan, Gonglin; Wei, Zengxin; Lu, Sha: Limited memory BFGS method with backtracking for symmetric nonlinear equations (2011)
  17. Andrei, Neculai: Accelerated scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization (2010)
  18. Chung, Julianne; Nagy, James G.; Sechopoulos, Ioannis: Numerical algorithms for polyenergetic digital breast tomosynthesis reconstruction (2010)
  19. Kim, Sunyoung; Kojima, Masakazu: Solving polynomial least squares problems via semidefinite programming relaxations (2010)
  20. Saad, Yousef; Chelikowsky, James R.; Shontz, Suzanne M.: Numerical methods for electronic structure calculations of materials (2010)

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