He, Chuan2023-11-302023-11-302023-07https://hdl.handle.net/11299/258868University of Minnesota Ph.D. dissertation. July 2023. Major: Industrial and Systems Engineering. Advisor: Zhaosong Lu. 1 computer file (PDF); vii, 195 pages.Second-order optimization has recently experienced significant developments, leading to numerous fruitful applications in science and engineering. In particular, recent research has shown that a second-order stationary point of a nonconvex optimization problem is often a globally optimal solution for instances that arise in areas such as machine learning and statistics. Therefore, developing efficient algorithms for computing such points is pivotal for advancing those areas. This dissertation introduces new algorithms with substantial theoretical improvements for nonconvex optimization problems and conducts numerical studies to show the practical advantages of the proposed methods over the state-of-the-art methods.enComplexity theoryNewton's methodNonconvex optimizationThesis or Dissertation