Wasserstein Stability and Free Boundaries in Measure-Parameterized Bilevel Obstacle Problems
Abstract
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 W11/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 ℝk, 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.
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BibTeX
@misc{huang2026wasserstein,
title = {{Wasserstein Stability and Free Boundaries in Measure-Parameterized Bilevel Obstacle Problems}},
author = {Kun Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08454},
eprint = {2609.08454},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08454},
}
RIS
TY - UNPB
TI - Wasserstein Stability and Free Boundaries in Measure-Parameterized Bilevel Obstacle Problems
AU - Kun Huang
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08454
UR - https://arxiv.org/abs/2609.08454
AB - 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 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.
ER -