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. Barbero, Álvaro; Sra, Suvrit: Modular proximal optimization for multidimensional total-variation regularization (2018)
  3. Beck, Amir; Vaisbourd, Yakov: Globally solving the trust region subproblem using simple first-order methods (2018)
  4. Bruins, Marianne; Duffy, James A.; Keane, Michael P.; Smith, Anthony A. jun.: Generalized indirect inference for discrete choice models (2018)
  5. Curtis, Frank E.; Robinson, Daniel P.; Samadi, Mohammadreza: Complexity analysis of a trust funnel algorithm for equality constrained optimization (2018)
  6. Dussault, Jean-Pierre: ARC$_q$: a new adaptive regularization by cubics (2018)
  7. Guan, Yu; Chu, Moody T.; Chu, Delin: SVD-based algorithms for the best rank-1 approximation of a symmetric tensor (2018)
  8. Guan, Yu; Chu, Moody T.; Chu, Delin: Convergence analysis of an SVD-based algorithm for the best rank-1 tensor approximation (2018)
  9. Jarre, Florian; Lieder, Felix: The solution of Euclidean norm trust region SQP subproblems via second-order cone programs: an overview and elementary introduction (2018)
  10. Jiang, Rujun; Li, Duan; Wu, Baiyi: SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices (2018)
  11. Khan, Kamil A.; Larson, Jeffrey; Wild, Stefan M.: Manifold sampling for optimization of nonconvex functions that are piecewise linear compositions of smooth components (2018)
  12. Lenders, Felix; Kirches, C.; Potschka, A.: trlib: a vector-free implementation of the GLTR method for iterative solution of the trust region problem (2018)
  13. Salahi, M.; Taati, A.: An efficient algorithm for solving the generalized trust region subproblem (2018)
  14. Salhov, Moshe; Bermanis, Amit; Wolf, Guy; Averbuch, Amir: Diffusion representations (2018)
  15. Sun, Ju; Qu, Qing; Wright, John: A geometric analysis of phase retrieval (2018)
  16. Yamada, Shinji; Takeda, Akiko: Successive Lagrangian relaxation algorithm for nonconvex quadratic optimization (2018)
  17. Zhang, Hao; Ni, Qin: A new regularized quasi-Newton method for unconstrained optimization (2018)
  18. Zhang, Lei-Hong; Shen, Chungen: A nested Lanczos method for the trust-region subproblem (2018)
  19. Zhang, Lei-Hong; Shen, Chungen; Yang, Wei Hong; Júdice, Joaquim J.: A Lanczos method for large-scale extreme Lorentz eigenvalue problems (2018)
  20. Zhang, Leihong; Yang, Weihong; Shen, Chungen; Feng, Jiang: Error bounds of Lanczos approach for trust-region subproblem (2018)

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