| Summary: | The Trust Region Subproblem (TRS) is the problem of minimizing a quadratic (possibly non-convex) function over a sphere. It is the main step of the trust region method for unconstrained optimization problems. Two cases may cause numerical difficulties in solving the TRS, i.e., (i) the so-called hard case and (ii) having a large trust region radius. In this thesis we give the optimality characteristics of the TRS and review the major current algorithms. Then we introduce some techniques to solve the TRS efficiently for the two difficult cases. A shift and deflation technique avoids the hard case; and a scaling can adjust the value of the trust region radius. In addition, we illustrate other improvements for the TRS algorithm, including: rotation, approximate eigenvalue calculations, and inverse polynomial interpolation. We also introduce a warm start approach and include a new treatment for the hard case for the trust region method. Sensitivity analysis is provided to show that the optimal objective value for the TRS is stable with respect to the trust region radius in both the easy and hard cases. Finally, numerical experiments are provided to show the performance of all the improvements.
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