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. Amaioua, Nadir; Audet, Charles; Conn, Andrew R.; Le Digabel, Sébastien: Efficient solution of quadratically constrained quadratic subproblems within the mesh adaptive direct search algorithm (2018)
  2. Beck, Amir; Vaisbourd, Yakov: Globally solving the trust region subproblem using simple first-order methods (2018)
  3. Bruins, Marianne; Duffy, James A.; Keane, Michael P.; Smith, Anthony A. jun.: Generalized indirect inference for discrete choice models (2018)
  4. Curtis, Frank E.; Robinson, Daniel P.; Samadi, Mohammadreza: Complexity analysis of a trust funnel algorithm for equality constrained optimization (2018)
  5. Dussault, Jean-Pierre: ARC$_q$: a new adaptive regularization by cubics (2018)
  6. Jarre, Florian; Lieder, Felix: The solution of Euclidean norm trust region SQP subproblems via second-order cone programs: an overview and elementary introduction (2018)
  7. Jiang, Rujun; Li, Duan; Wu, Baiyi: SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices (2018)
  8. Lenders, Felix; Kirches, C.; Potschka, A.: trlib: a vector-free implementation of the GLTR method for iterative solution of the trust region problem (2018)
  9. Zhang, Lei-Hong; Shen, Chungen; Yang, Wei Hong; Júdice, Joaquim J.: A Lanczos method for large-scale extreme Lorentz eigenvalue problems (2018)
  10. Adachi, Satoru; Iwata, Satoru; Nakatsukasa, Yuji; Takeda, Akiko: Solving the trust-region subproblem by a generalized eigenvalue problem (2017)
  11. Alkilayh, Maged; Reichel, Lothar; Yuan, Jin Yun: New zero-finders for trust-region computations (2017)
  12. Birgin, E. G.; Martínez, J. M.: The use of quadratic regularization with a cubic descent condition for unconstrained optimization (2017)
  13. Brust, Johannes; Erway, Jennifer B.; Marcia, Roummel F.: On solving L-SR1 trust-region subproblems (2017)
  14. Burdakov, Oleg; Gong, Lujin; Zikrin, Spartak; Yuan, Ya-xiang: On efficiently combining limited-memory and trust-region techniques (2017)
  15. Burer, Samuel; Kılınç-Karzan, Fatma: How to convexify the intersection of a second order cone and a nonconvex quadratic (2017)
  16. Curtis, Frank E.; Robinson, Daniel P.; Samadi, Mohammadreza: A trust region algorithm with a worst-case iteration complexity of $\mathcalO(\epsilon ^-3/2)$ for nonconvex optimization (2017)
  17. Ho-Nguyen, Nam; Kilinç-Karzan, Fatma: A second-order cone based approach for solving the trust-region subproblem and its variants (2017)
  18. Salahi, Maziar; Taati, Akram; Wolkowicz, Henry: Local nonglobal minima for solving large-scale extended trust-region subproblems (2017)
  19. Wang, Jiulin; Xia, Yong: A linear-time algorithm for the trust region subproblem based on hidden convexity (2017)
  20. Zhang, Lei-Hong; Shen, Chungen; Li, Ren-Cang: On the generalized Lanczos trust-region method (2017)

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