Dambrine Marc, Khan Akhtar A, Sama Miguel
Laboratoire de Mathématiques et de leurs Applications, Université de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LMAP, 64013 Pau Cedex, France.
School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USA.
Philos Trans A Math Phys Eng Sci. 2022 Nov 14;380(2236):20210352. doi: 10.1098/rsta.2021.0352. Epub 2022 Sep 26.
Numerous applied models used in the study of optimal control problems, inverse problems, shape optimization, machine learning, fractional programming, neural networks, image registration and so on lead to stochastic optimization problems in Hilbert spaces. Under a suitable convexity assumption on the objective function, a necessary and sufficient optimality condition for stochastic optimization problems is a stochastic variational inequality. This article presents a new stochastic regularized second-order iterative scheme for solving a variational inequality in a stochastic environment where the primary operator is accessed by employing sampling techniques. The proposed iterative scheme, which fits within the general framework of the stochastic approximation approach, has its almost-sure convergence analysis given in a Hilbert space. We test the feasibility and the efficacy of the proposed stochastic approximation approach for a stochastic optimal control problem and a stochastic inverse problem, both associated with a second-order stochastic partial differential equation. This article is part of the theme issue 'Non-smooth variational problems and applications'.
许多应用于最优控制问题、反问题、形状优化、机器学习、分式规划、神经网络、图像配准等研究中的模型都会导致希尔伯特空间中的随机优化问题。在目标函数的适当凸性假设下，随机优化问题的一个充要最优性条件是一个随机变分不等式。本文提出了一种新的随机正则化二阶迭代格式，用于求解随机环境下的变分不等式，其中主要算子是通过采样技术获取的。所提出的迭代格式符合随机逼近方法的一般框架，其几乎必然收敛性分析是在希尔伯特空间中给出的。我们针对与二阶随机偏微分方程相关的一个随机最优控制问题和一个随机反问题，测试了所提出的随机逼近方法的可行性和有效性。本文是“非光滑变分问题及应用”主题专刊的一部分。
