【Team】Continuous Optimization Team
【Date】2026/July/7(Tuesday) 16:00-17:00(JST)
【Speaker】Yuntian Jiang, Shanghai University of Finance and Economics
Title: A New Trust Region Method with Advanced Complexity Analyses
Abstract: The trust-region (TR) method is renowned for its numerical robustness, yet its theoretical guarantees in convex optimization and global acceleration remain underexplored. This talk addresses these gaps through two connected works.
First, I will introduce a universal TR framework that simultaneously incorporates quadratic regularization and ball constraints. By developing a novel descent property, we unify the analysis for both convex and nonconvex optimization. This universal method achieves an iteration complexity of $\tilde{O}(\epsilon^{-1/2})$ for convex problems, and $\tilde{O}(\epsilon^{-3/2})$ to find an approximate second-order stationary point in nonconvex settings.
Second, building upon this foundation, we tackle the fundamental trade-off between global efficiency and fast local convergence in second-order methods. We propose the first accelerated TR-type methods by leveraging inherent primal-dual information. Specifically, our ``Accelerating with Local Detection'' approach achieves a global oracle complexity of $\tilde{O}(\epsilon^{-1/3})$ while strictly maintaining quadratic local convergence. Furthermore, we establish a theoretical phase transition: pushing global efficiency to the limit ($\tilde{O}(\epsilon^{-2/7})$) inherently breaks down quadratic local convergence. Together, these works provide a comprehensive understanding of the theoretical limits and practical capabilities of modern trust-region methods.
