A Homogeneous Second-Order Descent Method for Nonconvex Optimization

• 成果类型: Article; Early Access
• 署名作者: Zhang, Chuwen; He, Chang; Jiang, Yuntian; Xue, Chenyu; Jiang, Bo; Ge, Dongdong; Ye, Yinyu
• 署名单位: Shanghai University of Finance & Economics; Shanghai University of Finance & Economics; Shanghai University of Finance & Economics; Shanghai Jiao Tong University; Stanford University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2023.0132
• 发表日期: 2025
• 关键词: cubic regularization; Newton method; complexity; EIGENVALUE; subproblem; algorithms; cg

摘要

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 O(epsilon-3=2) to find an epsilon-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.