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. Zhang, Lei-Hong; Shen, Chungen; Yang, Wei Hong; Júdice, Joaquim J.: A Lanczos method for large-scale extreme Lorentz eigenvalue problems (2018)
  2. Adachi, Satoru; Iwata, Satoru; Nakatsukasa, Yuji; Takeda, Akiko: Solving the trust-region subproblem by a generalized eigenvalue problem (2017)
  3. Alkilayh, Maged; Reichel, Lothar; Yuan, Jin Yun: New zero-finders for trust-region computations (2017)
  4. Birgin, E.G.; Martínez, J.M.: The use of quadratic regularization with a cubic descent condition for unconstrained optimization (2017)
  5. Brust, Johannes; Erway, Jennifer B.; Marcia, Roummel F.: On solving L-SR1 trust-region subproblems (2017)
  6. Burdakov, Oleg; Gong, Lujin; Zikrin, Spartak; Yuan, Ya-xiang: On efficiently combining limited-memory and trust-region techniques (2017)
  7. Burer, Samuel; Kılınç-Karzan, Fatma: How to convexify the intersection of a second order cone and a nonconvex quadratic (2017)
  8. 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)
  9. Ho-Nguyen, Nam; Kilinç-Karzan, Fatma: A second-order cone based approach for solving the trust-region subproblem and its variants (2017)
  10. Salahi, Maziar; Taati, Akram; Wolkowicz, Henry: Local nonglobal minima for solving large-scale extended trust-region subproblems (2017)
  11. Wang, Jiulin; Xia, Yong: A linear-time algorithm for the trust region subproblem based on hidden convexity (2017)
  12. Zhang, Lei-Hong; Shen, Chungen; Li, Ren-Cang: On the generalized Lanczos trust-region method (2017)
  13. Ahookhosh, Masoud; Ghaderi, Susan: Two globally convergent nonmonotone trust-region methods for unconstrained optimization (2016)
  14. Beck, Amir; Hallak, Nadav: On the minimization over sparse symmetric sets: projections, optimality conditions, and algorithms (2016)
  15. Bernal, Francisco: Trust-region methods for nonlinear elliptic equations with radial basis functions (2016)
  16. Hazan, Elad; Koren, Tomer: A linear-time algorithm for trust region problems (2016)
  17. Jiang, Rujun; Li, Duan: Simultaneous diagonalization of matrices and its applications in quadratically constrained quadratic programming (2016)
  18. Minoux, Michel; Zorgati, Riadh: Convexity of Gaussian chance constraints and of related probability maximization problems (2016)
  19. Shtern, Shimrit; Ben-Tal, Aharon: Computational methods for solving nonconvex block-separable constrained quadratic problems (2016)
  20. Yang, Boshi; Burer, Samuel: A two-variable approach to the two-trust-region subproblem (2016)

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