Computing a trust region step We propose an algorithm for the problem of minimizing a quadratic function subject to an ellipsoidal constraint and show that this algorithm is guaranteed to produce a nearly optimal solution in a finite number of iterations. We also consider the use of this algorithm in a trust region Newton’s method. In particular, we prove that under reasonable assumptions the sequence generated by Newton’s method has a limit point which satisfies the first and second order necessary conditions for a minimizer of the objective function. Numerical results for GQTPAR, which is a Fortran implementation of our algorithm, show that GQTPAR is quite successful in a trust region method. In our tests a call to GQTPAR only required 1.6 iterations on the average.

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  1. Curtis, Frank E.; Robinson, Daniel P.; Royer, Clément W.; Wright, Stephen J.: Trust-region Newton-CG with strong second-order complexity guarantees for nonconvex optimization (2021)
  2. Hoffmann, Alexandre; Monteiller, Vadim; Bellis, Cédric: A penalty-free approach to PDE constrained optimization: application to an inverse wave problem (2021)
  3. Jia, Zhongxiao; Wang, Fa: The convergence of the generalized Lanczos trust-region method for the trust-region subproblem (2021)
  4. Kanzow, Christian; Lechner, Theresa: Globalized inexact proximal Newton-type methods for nonconvex composite functions (2021)
  5. Moosaei, Hossein; Hladík, Milan: On the optimal correction of infeasible systems of linear inequalities (2021)
  6. Wang, Rui; Xiu, Naihua; Toh, Kim-Chuan: Subspace quadratic regularization method for group sparse multinomial logistic regression (2021)
  7. Wang, Rui; Xiu, Naihua; Zhou, Shenglong: An extended Newton-type algorithm for (\ell_2)-regularized sparse logistic regression and its efficiency for classifying large-scale datasets (2021)
  8. Bahrami, Somayeh; Amini, Keyvan: An efficient two-step trust-region algorithm for exactly determined consistent systems of nonlinear equations (2020)
  9. Brás, C. P.; Martínez, J. M.; Raydan, M.: Large-scale unconstrained optimization using separable cubic modeling and matrix-free subspace minimization (2020)
  10. Cartis, Coralia; Gould, Nicholas I. M.; Lange, Marius: On monotonic estimates of the norm of the minimizers of regularized quadratic functions in Krylov spaces (2020)
  11. Costa, Carina Moreira; Grapiglia, Geovani Nunes: A subspace version of the Wang-Yuan augmented Lagrangian-trust region method for equality constrained optimization (2020)
  12. Daneshmand, Amir; Scutari, Gesualdo; Kungurtsev, Vyacheslav: Second-order guarantees of distributed gradient algorithms (2020)
  13. Erway, Jennifer B.; Griffin, Joshua; Marcia, Roummel F.; Omheni, Riadh: Trust-region algorithms for training responses: machine learning methods using indefinite Hessian approximations (2020)
  14. Gao, Guohua; Jiang, Hao; Vink, Jeroen C.; van Hagen, Paul P. H.; Wells, Terence J.: Performance enhancement of Gauss-Newton trust-region solver for distributed Gauss-Newton optimization method (2020)
  15. Gould, Nicholas I. M.; Simoncini, Valeria: Error estimates for iterative algorithms for minimizing regularized quadratic subproblems (2020)
  16. Jiang, Rujun; Li, Duan: On conic relaxations of generalization of the extended trust region subproblem (2020)
  17. Jiang, Rujun; Li, Duan: A linear-time algorithm for generalized trust region subproblems (2020)
  18. Lieder, Felix: Solving large-scale cubic regularization by a generalized eigenvalue problem (2020)
  19. Milz, Johannes; Ulbrich, Michael: An approximation scheme for distributionally robust nonlinear optimization (2020)
  20. 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)

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