Complexity and global rates of trust-region methods based on probabilistic models

Trust-region algorithms have been proved to globally converge with probability 1 when the accuracy of the trust-region models is imposed with a certain probability conditioning on the iteration history. In this article, we study the complexity of such methods, providing global rates and worst-case complexity bounds on the number of iterations (with overwhelmingly high probability), for both first- and second-order measures of optimality. Such results are essentially the same as the ones known for trust-region methods based on deterministic models. The derivation of the global rates and worst-case complexity bounds follows closely from a study of direct search methods based on the companion notion of probabilistic descent.