In this paper we study the limit of the value function for a two-scale, infinite-dimensional, stochastic controlled system with cylindrical noise and possibly degenerate diffusion. The limit is represented as the value function of a new reduced control problem (on a reduced state space). The presence of a cylindrical noise prevents representation of the limit by viscosity solutions of HJB equations, while degeneracy of diffusion coefficients prevents representation as a classical BSDE. We use a vanishing noise regularization technique.
A Study of One-Parameter Regularization Methods for Mathematical Programs with Vanishing Constraints
带消失约束的数学规划（Mathematical Programmes with Vanishing Constraints，MPVCs）是一类非线性最优化问题，在桁架拓扑设计、机器人运动规划等工程问题中有着广泛的应用。从理论和数值的角度来看，MPVCs都是困难的问题：消失约束的组合性质通常会阻止标准的约束规格和最优性条件的实现;此外，可行集本质上是非凸的，并且通常在感兴趣的点周围没有内部。因此，本文研究和比较了四种正则化方法的数值解的MPH。每种方法都依赖于一个正则化参数，该参数用于将原始MPVC嵌入到标准非线性程序序列中。这些方法的收敛结果的基础上精确和近似平稳的子问题建立在弱假设。通过给出KKT乘子存在的充分条件，研究了子问题的改进正则性。基于桁架拓扑设计和气动热力学最优控制问题的数值实验，补充了正则化方法的理论分析和比较。计算结果突出了使用正则化直接应用标准求解器的好处，它们使我们能够确定两个有前途的正则化方案。
Tikhonov regularization of monotone operator flows not only ensures strong convergence of the trajectories but also speeds up the vanishing of the residuals
Fast convergence rates and trajectory convergence of a Tikhonov regularized inertial primal\mbox{-}dual dynamical system with time scaling and vanishing damping
Tikhonov regularized second-order plus first-order primal-dual dynamical systems with asymptotically vanishing damping for linear equality constrained convex optimization problems
Combining Strong Convergence, Values Fast Convergence and Vanishing of Gradients for a Proximal Point Algorithm Using Tikhonov Regularization in a Hilbert Space
Equilibrium under Time-Inconsistency: A New Existence Theory by Vanishing Entropy Regularization
Zhongtai Securities Institute for Financial Studies, Shandong University（山东大学中泰金融研究院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）；China School of Banking and Finance, University of International Business and Economics（对外经济贸易大学中国金融学院）；School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学院）
