Abstract: Following the development of a group of efficient derivative-free optimization methods by Michael J.D. Powell in the 2000s, a series of analytical work was conducted to provide theoretical guarantees to these algorithms. These analyses focus on the behavior of the trust-region method routine within Powell’s algorithms while treating the derivatives approximated via zeroth-order information as outputs of noisy oracles. Dr. Liyuan Cao presents his contributions to this line of work including an analysis on the common gradient approximation methods and his recent work on a modified trust-region method designed to handle noise in function evaluations.