测度参数化双层障碍问题中的Wasserstein稳定性与自由边界
Wasserstein Stability and Free Boundaries in Measure-Parameterized Bilevel Obstacle Problems
浏览论文内容
中文总结 AI 辅助
本文研究测度参数化双层障碍问题，通过凸性和Poincare不等式建立策略映射的Wasserstein稳定性，并分析自由边界的模量、经验律收敛速率及税收应用。
中文摘要 AI 辅助
我们研究受障碍约束的变分问题，其约化能量通过下层优化器依赖于概率律。一致强凸性给出单值Lipschitz跟随者响应，而凸性和Poincare不等式给出唯一的上层策略。类型Lipschitz约化边际则蕴含从1-Wasserstein度量到能量空间的策略映射的Lipschitz连续性。在一维线性障碍子类中，策略导数是累积强迫的正部。单交叉使重合集成为区间。m阶代数交叉给出其端点的$W_1^{1/m}$模量；奇次幂例子表明该指数是尖锐的，而横截交叉恢复Lipschitz稳定性。对于$\nmathbb{R}^k$紧子集上的经验律，维数相关的Wasserstein界给出策略和阈值的有限样本速率。在横截总体根处，经验阈值渐近线性，具有显式影响函数和中心极限定理。对于高维正则片，条件水平集论证在非退化切换函数可用时给出局部Hausdorff稳定性。显式二次坚定响应产生非线性公司税表，具有内生零税区域和Wasserstein稳定阈值。税收示例是解析的，不使用经验校准。
英文摘要
We study obstacle-constrained variational problems whose reduced energies depend on a probability law through a lower-level optimizer. Uniform strong convexity yields a single-valued Lipschitz follower response, while convexity and a Poincare inequality give a unique upper-level policy. A type-Lipschitz reduced marginal then implies Lipschitz continuity of the policy map from the 1-Wasserstein metric to the energy space. In a one-dimensional linear-obstacle subclass, the policy derivative is the positive part of a cumulative forcing. Single crossing makes the coincidence set an interval. An algebraic crossing of order m gives a $W_1^{1/m}$ modulus for its endpoint; odd-power examples show that this exponent is sharp, while a transversal crossing recovers Lipschitz stability. For empirical laws on compact subsets of $\mathbb{R}^k$, dimension-dependent Wasserstein bounds yield finite-sample rates for policies and thresholds. At a transversal population root, the empirical threshold is asymptotically linear, with an explicit influence function and central limit theorem. For higher-dimensional regular patches, a conditional level-set argument gives local Hausdorff stability when a nondegenerate switching function is available. An explicit quadratic firm response produces a nonlinear corporate-tax schedule with an endogenous zero-tax region and a Wasserstein-stable threshold. The tax illustration is analytic and uses no empirical calibration.
发表机构
- Economics and Management School, Wuhan University（武汉大学经济与管理学院）
机构由 AI 辅助整理，请以论文原文为准。
