改进了非凸优化中的信任域方法，支持更广泛的非光滑项。
An Inexact Weighted Proximal Trust-Region Method
- 用δ-Fréchet次微分扩展近似邻近算子定义
- 在加权内积下保证算法收敛性
- 成功求解含Burgers方程的最优控制问题
在[Baraldi and Kouri, Math. Program., 201:1 (2023), pp. 559-598]中，作者提出了一种用于最小化光滑非凸函数与具有解析邻近算子的非光滑凸函数之和的信任域方法。尽管许多函数满足此条件（如ℓ₁范数在ℓ₂空间上），但拓扑或非光滑项性质限制了其适用范围。本文利用δ-Fréchet次微分扩展了近似邻近算子的定义，使其可嵌入前述信任域算法。同时，将标准信任域收敛理论推广至处理加权内积下的邻近算子不精确性。首先设计生成近似邻近算子点的算法，随后将其应用于求解约束于Burgers方程的最优控制问题。
原文摘要 · Abstract (English)
In [R. J. Baraldi and D. P. Kouri, Math. Program., 201:1 (2023), pp. 559-598], the authors introduced a trust-region method for minimizing the sum of a smooth nonconvex and a nonsmooth convex function, the latter of which has an analytical proximity operator. While many functions satisfy this criterion, e.g., the $\ell_1$-norm defined on $\ell_2$, many others are precluded by either the topology or the nature of the nonsmooth term. Using the $δ$-Fréchet subdifferential, we extend the definition of the inexact proximity operator and enable its use within the aforementioned trust-region algorithm. Moreover, we augment the analysis for the standard trust-region convergence theory to handle proximity operator inexactness with weighted inner products. We first introduce an algorithm to generate a point in the inexact proximity operator and then apply the algorithm within the trust-region method to solve an optimal control problem constrained by Burgers' equation.
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