发表机构
Sandia National Laboratories; Oregon State University（桑迪亚国家实验室; 俄勒冈州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出并分析了一种非精确近端信赖域算法，用于求解光滑非凸与非光滑凸函数之和的最小化问题，通过允许近端算子非精确评估保证全局收敛，并在PDE约束优化中验证了其有效性。
AI 中文摘要
我们分析了一种非精确的近端信赖域算法，用于最小化光滑非凸函数与非光滑凸函数之和。该算法利用近端梯度作为基准以保证全局收敛。然而，评估近端梯度需要求解一个非光滑凸优化问题，在许多应用中该问题无法精确求解。因此，我们开发了一个允许近端算子非精确评估的框架，并证明了所提出的非精确性条件能产生与精确近端梯度类似的下降性质。我们进一步讨论了两种常见应用中计算近似近端算子的实用程序：（i）非光滑项为多个非光滑凸函数之和，其中一个与仿射映射复合；（ii）使用替代内积以促进近端算子的评估。我们通过两个源于PDE约束优化的数值实验展示了我们的算法并证实了我们的分析。
英文摘要
We analyze an inexact, proximal trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function. Our algorithm leverages the proximal gradient as a benchmark to guarantee global convergence. However, evaluating the proximal gradient involves solving a nonsmooth convex optimization problem that in many applications cannot be performed exactly. As such, we develop a framework that permits inexact evaluations of the proximity operator and we show the proposed inexactness conditions yield descent properties analogous to the exact proximal gradient. We further discuss practical procedures for computing approximate proximity operators for two common applications: (i) the nonsmooth term is the sum of nonsmooth convex functions where one is composed with an affine map and (ii) using alternative inner products to facilitate the evaluation of the proximity operator. We demonstrate our algorithm and confirm our analysis with two numerical experiments arising in PDE-constrained optimization.
