A Homogeneous Second-Order Descent Method for Nonconvex Optimization

In this paper, we introduce a homogeneous second-order descent method (HSODM) motivated from the homogenization trick in quadratic programming. The merit of homogenization is that only the leftmost eigenvector of a gradient-Hessian integrated matrix is computed at each iteration. Therefore, the algorithm is a single-loop method that does not need to switch to other sophisticated algorithms and is easy to implement. We show that HSODM has a global convergence rate of [Formula: see text] to find an [Formula: see text]-approximate second-order stationary point, and has a local quadratic convergence rate under the standard assumptions. The numerical results demonstrate the advantage of the proposed method over other second-order methods. Funding: This research was partially supported by the National Natural Science Foundation of China [Grants 72394360, 72394364, 72394365, 72225009, and 72171141] and by the Program for Innovative Research Team of the Shanghai University of Finance and Economics. Supplemental Material: Supplemental material is available at https://doi.org/10.1287/moor.2023.0132 .