Higher Order Adaptive Regularization Methods for Unconstrained Optimization

This thesis deals with the so called 'adaptive-regularization'(AR) methods. Such methods arise from the well-known trust-region(TR) methods, as an attempt to improve efficiency and attain faster convergence. Following in the footsteps of Coralia Cartis, the first part provides an overview of the methods, with a full complexity analysis and an introduction to solving the minimization sub-problem, which can perhaps be seen as the crucial step of the algorithms. On the other hand, the last chapter is all about AR3, i.e. third-order adaptive regularization method; for which a few numerical tests are carried out.