In Baraldi (Math Program 20:1–40, 2022), we introduced an inexact trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function in Hilbert space—a class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. This algorithm has demonstrated excellent performance and scalability with problem size. In this paper, we enrich the convergence analysis for this algorithm, proving strong convergence of the iterates with guaranteed rates. In particular, we demonstrate that the trust-region algorithm recovers superlinear, even quadratic, convergence rates when using a second-order Taylor approximation of the smooth objective function term.

在 Baraldi（Math Program 20:1–40, 2022）中，我们引入了一种不精确的信赖域算法，用于最小化希尔伯特空间中的平滑非凸函数和非平滑凸函数的总和，这是数据科学中普遍存在的一类问题、学习、最优控制和逆问题。该算法展示了出色的性能和问题规模的可扩展性。在本文中，我们丰富了该算法的收敛分析，证明了在保证速率的情况下迭代的强收敛性。特别是，我们证明，当使用平滑目标函数项的二阶泰勒近似时，信任域算法可以恢复超线性甚至二次收敛率。