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. Adachi, Satoru; Iwata, Satoru; Nakatsukasa, Yuji; Takeda, Akiko: Solving the trust-region subproblem by a generalized eigenvalue problem (2017)
  2. Birgin, E.G.; Martínez, J.M.: The use of quadratic regularization with a cubic descent condition for unconstrained optimization (2017)
  3. Ho-Nguyen, Nam; Kilinç-Karzan, Fatma: A second-order cone based approach for solving the trust-region subproblem and its variants (2017)
  4. Ahookhosh, Masoud; Ghaderi, Susan: Two globally convergent nonmonotone trust-region methods for unconstrained optimization (2016)
  5. Beck, Amir; Hallak, Nadav: On the minimization over sparse symmetric sets: projections, optimality conditions, and algorithms (2016)
  6. Hazan, Elad; Koren, Tomer: A linear-time algorithm for trust region problems (2016)
  7. Jiang, Rujun; Li, Duan: Simultaneous diagonalization of matrices and its applications in quadratically constrained quadratic programming (2016)
  8. Minoux, Michel; Zorgati, Riadh: Convexity of Gaussian chance constraints and of related probability maximization problems (2016)
  9. Shtern, Shimrit; Ben-Tal, Aharon: Computational methods for solving nonconvex block-separable constrained quadratic problems (2016)
  10. Yang, Boshi; Burer, Samuel: A two-variable approach to the two-trust-region subproblem (2016)
  11. Ben-Tal, Aharon; Hazan, Elad; Koren, Tomer; Mannor, Shie: Oracle-based robust optimization via online learning (2015)
  12. Burer, Samuel; Yang, Boshi: The trust region subproblem with non-intersecting linear constraints (2015)
  13. Erway, Jennifer B.; Marcia, Roummel F.: On efficiently computing the eigenvalues of limited-memory quasi-Newton matrices (2015)
  14. Hicken, Jason E.; Dener, Alp: A flexible iterative solver for nonconvex, equality-constrained quadratic subproblems (2015)
  15. Jin, Ping; Ling, Chen; Shen, Huifei: A smoothing Levenberg-Marquardt algorithm for semi-infinite programming (2015)
  16. Martínez, J.M.; Raydan, M.: Separable cubic modeling and a trust-region strategy for unconstrained minimization with impact in global optimization (2015)
  17. Powell, M.J.D.: On fast trust region methods for quadratic models with linear constraints (2015)
  18. Wang, Shu; Xia, Yong: Strong duality for generalized trust region subproblem: S-lemma with interval bounds (2015)
  19. Pong, Ting Kei; Wolkowicz, Henry: The generalized trust region subproblem (2014)
  20. Pulkkinen, Seppo; Mäkelä, Marko M.; Karmitsa, Napsu: A generative model and a generalized trust region Newton method for noise reduction (2014)

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