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. Aravkin, Aleksandr Y.; Baraldi, Robert; Orban, Dominique: A proximal quasi-Newton trust-region method for nonsmooth regularized optimization (2022)
  2. Jia, Xiaojing; Liang, Xin; Shen, Chungen; Zhang, Lei-Hong: Solving the cubic regularization model by a nested restarting Lanczos method (2022)
  3. Karbasy, Saeid Ansary; Salahi, Maziar: On the branch and bound algorithm for the extended trust-region subproblem (2022)
  4. Li, Bin; Lei, Yuan: Hybrid algorithms with active set prediction for solving linear inequalities in a least squares sense (2022)
  5. Lin, Matthew M.; Chu, Moody T.: Rank-1 approximation for entangled multipartite real systems (2022)
  6. Ma, Xijun; Shen, Chungen; Wang, Li; Zhang, Lei-Hong; Li, Ren-Cang: A self-consistent-field iteration for MAXBET with an application to multi-view feature extraction (2022)
  7. Milz, Johannes; Ulbrich, Michael: An approximation scheme for distributionally robust PDE-constrained optimization (2022)
  8. Wang, Alex L.; Kılınç-Karzan, Fatma: The generalized trust region subproblem: solution complexity and convex hull results (2022)
  9. Zeng, Liaoyuan; Pong, Ting Kei: (\rho)-regularization subproblems: strong duality and an eigensolver-based algorithm (2022)
  10. 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)
  11. Hoffmann, Alexandre; Monteiller, Vadim; Bellis, Cédric: A penalty-free approach to PDE constrained optimization: application to an inverse wave problem (2021)
  12. Jia, Zhongxiao; Wang, Fa: The convergence of the generalized Lanczos trust-region method for the trust-region subproblem (2021)
  13. Kanzow, Christian; Lechner, Theresa: Globalized inexact proximal Newton-type methods for nonconvex composite functions (2021)
  14. Larson, Jeffrey; Menickelly, Matt; Zhou, Baoyu: Manifold sampling for optimizing nonsmooth nonconvex compositions (2021)
  15. Moosaei, Hossein; Hladík, Milan: On the optimal correction of infeasible systems of linear inequalities (2021)
  16. Wang, Rui; Xiu, Naihua; Toh, Kim-Chuan: Subspace quadratic regularization method for group sparse multinomial logistic regression (2021)
  17. 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)
  18. Zhou, Shenglong; Pan, Lili; Xiu, Naihua: Newton method for (\ell_0)-regularized optimization (2021)
  19. Bahrami, Somayeh; Amini, Keyvan: An efficient two-step trust-region algorithm for exactly determined consistent systems of nonlinear equations (2020)
  20. Brás, C. P.; Martínez, J. M.; Raydan, M.: Large-scale unconstrained optimization using separable cubic modeling and matrix-free subspace minimization (2020)

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