The choice of the step in trust region methods

The main goal of this thesis is the choice of steps in trust region methods for finding a minimum of a given function. The step corresponds to the problem of finding a minimum of a model function on a trust region. We characterize a solu- tion of this problem (Moré-Sorensen theorem) and consider various techniques for approximating a solution of this problem (the Cauchy point method, the dogleg method, the conjugate gradients method). In the case of the first two techniques we prove convergence of the optimization method. Finally, the above techniques are tested numerically in MATLAB on properly chosen functions and initial data. We comment on advantages and disadvantages of considered algorithms.