AI 中文总结
本文开发数值分裂技术，提出多尺度惯性前向后向分裂算法，用于解决约束单调包含问题，能处理多种惩罚算子，建立了迭代弱收敛性，在特定条件下证明了向最小范数解的强收敛性。
AI 中文摘要
在实希尔伯特空间环境下，本文开发了数值分裂技术，以确保强收敛到约束变分不等式的最小范数解。我们提出一种多尺度惯性前向后向分裂算法，用于解决具有多尺度惩罚和消失蒂霍诺夫正则化的约束单调包含问题。该框架适用于光滑、非光滑和混合光滑 - 非光滑惩罚算子，统一处理了一大类约束单调包含问题。在此框架下，我们建立了生成迭代的弱收敛性。通过引入离散蒂霍诺夫中心路径，在关于问题数据的温和约束限定条件下，进一步证明了向问题最小范数解的强收敛性。
英文摘要
In a real Hilbertian setting, we develop in this paper numerical splitting techniques guaranteeing strong convergence to the least norm solution of constrained variational inequalities. We develop a multiscale inertial forward-backward splitting algorithm for solving constrained monotone inclusion problems with multiscale penalization and vanishing Tikhonov regularization. The proposed framework accommodates smooth, nonsmooth, and mixed smooth--nonsmooth penalty operators, providing a unified treatment of a broad class of constrained monotone inclusion problems. In this general framework, we establish weak convergence of the generated iterates. By introducing a discrete Tikhonov central path, we further prove strong convergence to the minimum-norm solution of the problem under a mild constraint qualification condition on the problem data.
