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. Bahrami, Somayeh; Amini, Keyvan: An efficient two-step trust-region algorithm for exactly determined consistent systems of nonlinear equations (2020)
  2. Brás, C. P.; Martínez, J. M.; Raydan, M.: Large-scale unconstrained optimization using separable cubic modeling and matrix-free subspace minimization (2020)
  3. Erway, Jennifer B.; Griffin, Joshua; Marcia, Roummel F.; Omheni, Riadh: Trust-region algorithms for training responses: machine learning methods using indefinite Hessian approximations (2020)
  4. 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)
  5. Gould, Nicholas I. M.; Simoncini, Valeria: Error estimates for iterative algorithms for minimizing regularized quadratic subproblems (2020)
  6. Jiang, Rujun; Li, Duan: On conic relaxations of generalization of the extended trust region subproblem (2020)
  7. 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)
  8. Taati, Akram; Salahi, Maziar: On local non-global minimizers of quadratic optimization problem with a single quadratic constraint (2020)
  9. Wang, Li-Yan; Liu, Ji-Jun: On fluorophore imaging by diffusion equation model: decompositions and optimizations (2020)
  10. Adachi, Satoru; Nakatsukasa, Yuji: Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint (2019)
  11. Amiri, Erfan A.; Craig, James R.; Hirmand, M. Reza: A trust region approach for numerical modeling of non-isothermal phase change (2019)
  12. Birgin, E. G.; Martínez, J. M.: A Newton-like method with mixed factorizations and cubic regularization for unconstrained minimization (2019)
  13. Brust, Johannes J.; Marcia, Roummel F.; Petra, Cosmin G.: Large-scale quasi-Newton trust-region methods with low-dimensional linear equality constraints (2019)
  14. Guan, Yu; Chu, Delin: Numerical computation for orthogonal low-rank approximation of tensors (2019)
  15. Huang, Baohua; Ma, Changfeng: The least squares solution of a class of generalized Sylvester-transpose matrix equations with the norm inequality constraint (2019)
  16. Jiang, Rujun; Li, Duan: Novel reformulations and efficient algorithms for the generalized trust region subproblem (2019)
  17. Larson, Jeffrey; Menickelly, Matt; Wild, Stefan M.: Derivative-free optimization methods (2019)
  18. Lee, Ching-pei; Wright, Stephen J.: Inexact successive quadratic approximation for regularized optimization (2019)
  19. Nino-Ruiz, Elias D.; Ardila, Carlos; Estrada, Jesus; Capacho, Jose: A reduced-space line-search method for unconstrained optimization via random descent directions (2019)
  20. Paternain, Santiago; Mokhtari, Aryan; Ribeiro, Alejandro: A Newton-based method for nonconvex optimization with fast evasion of saddle points (2019)

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