Solving the trust-region subproblem using the Lanczos method. The approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods. When the number of variables is large, the most widely used strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trust-region boundary. We investigate ways of continuing the process once the boundary has been encountered. The key is to observe that the trust-region problem within the currently generated Krylov subspace has a very special structure which enables it to be solved very efficiently. We compare the new strategy with existing methods. The resulting software package is available as HSL-VF05 within the Harwell Subroutine Library.

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  1. Brás, C. P.; Martínez, J. M.; Raydan, M.: Large-scale unconstrained optimization using separable cubic modeling and matrix-free subspace minimization (2020)
  2. Carmon, Yair; Duchi, John C.: First-order methods for nonconvex quadratic minimization (2020)
  3. Chieu, Nguyen Huy; Van Hien, Le; Trang, Nguyen Thi Quynh: Tilt stability for quadratic programs with one or two quadratic inequality constraints (2020)
  4. Erway, Jennifer B.; Griffin, Joshua; Marcia, Roummel F.; Omheni, Riadh: Trust-region algorithms for training responses: machine learning methods using indefinite Hessian approximations (2020)
  5. Gould, Nicholas I. M.; Simoncini, Valeria: Error estimates for iterative algorithms for minimizing regularized quadratic subproblems (2020)
  6. Nguyen, Van-Bong; Nguyen, Thi Ngan; Sheu, Ruey-Lin: Strong duality in minimizing a quadratic form subject to two homogeneous quadratic inequalities over the unit sphere (2020)
  7. Taati, Akram; Salahi, Maziar: On local non-global minimizers of quadratic optimization problem with a single quadratic constraint (2020)
  8. Wang, Jiulin; Xia, Yong: Closing the gap between necessary and sufficient conditions for local nonglobal minimizer of trust region subproblem (2020)
  9. Xu, Peng; Roosta, Fred; Mahoney, Michael W.: Newton-type methods for non-convex optimization under inexact Hessian information (2020)
  10. Adachi, Satoru; Nakatsukasa, Yuji: Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint (2019)
  11. Carmon, Yair; Duchi, John: Gradient descent finds the cubic-regularized nonconvex Newton step (2019)
  12. Furini, Fabio; Traversi, Emiliano; Belotti, Pietro; Frangioni, Antonio; Gleixner, Ambros; Gould, Nick; Liberti, Leo; Lodi, Andrea; Misener, Ruth; Mittelmann, Hans; Sahinidis, Nikolaos V.; Vigerske, Stefan; Wiegele, Angelika: QPLIB: a library of quadratic programming instances (2019)
  13. Huang, Baohua; Ma, Changfeng: The least squares solution of a class of generalized Sylvester-transpose matrix equations with the norm inequality constraint (2019)
  14. Song, Liqiang; Yang, Wei Hong: A block Lanczos method for the extended trust-region subproblem (2019)
  15. Taati, A.; Salahi, M.: A conjugate gradient-based algorithm for large-scale quadratic programming problem with one quadratic constraint (2019)
  16. Zhou, W.; Akrotirianakis, I. G.; Yektamaram, S.; Griffin, J. D.: A matrix-free line-search algorithm for nonconvex optimization (2019)
  17. Beck, Amir; Vaisbourd, Yakov: Globally solving the trust region subproblem using simple first-order methods (2018)
  18. Lenders, Felix; Kirches, C.; Potschka, A.: \texttttrlib: a vector-free implementation of the GLTR method for iterative solution of the trust region problem (2018)
  19. Salahi, M.; Taati, A.: An efficient algorithm for solving the generalized trust region subproblem (2018)
  20. Salahi, M.; Taati, A.: A fast eigenvalue approach for solving the trust region subproblem with an additional linear inequality (2018)

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