Analysis of Bregman Proximal Point for the Obstacle Problem
Abstract
We study the Bregman proximal point method for the obstacle problem, a fundamental variational inequality arising in contact mechanics, optimal design, and mathematical finance. Each Bregman proximal step regularizes the energy through a Bregman divergence generated by a Legendre function, leading to a semilinear elliptic subproblem. We first establish a well-posedness and strict feasibility theory for these subproblems. We then analyze the convergence of the resulting iteration in the H1-norm. Our convergence analysis is based on a sequential strict local minimality inequality. We show abstractly how the order of minimality determines the convergence rate. Assuming that the initial guess lies above the exact solution and a one-sided Bregman growth condition holds, we establish sequential strict local minimality of order s ≥ 2 and derive sublinear convergence rates in the H1-norm of the form O(k−ζ), where ζ∈(1/2,1] depends on the choice of Legendre function. For the Shannon and Tsallis entropies, we show that these rates are sharp in a uniform worst-case sense. Under additional assumptions on the free boundary and the obstacle, we establish sequential strict local minimality of order s = 1 with the Shannon and Spence entropies. This yields linear convergence of the form O(ϱk) for some ϱ∈(0,1).
Related
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BibTeX
@misc{keith2026analysis,
title = {{Analysis of Bregman Proximal Point for the Obstacle Problem}},
author = {Brendan Keith and Haojun Qin and Noe Reyes Rivas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23007},
eprint = {2609.23007},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23007},
}
RIS
TY - UNPB
TI - Analysis of Bregman Proximal Point for the Obstacle Problem
AU - Brendan Keith
AU - Haojun Qin
AU - Noe Reyes Rivas
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23007
UR - https://arxiv.org/abs/2609.23007
AB - We study the Bregman proximal point method for the obstacle problem, a fundamental variational inequality arising in contact mechanics, optimal design, and mathematical finance. Each Bregman proximal step regularizes the energy through a Bregman divergence generated by a Legendre function, leading to a semilinear elliptic subproblem. We first establish a well-posedness and strict feasibility theory for these subproblems. We then analyze the convergence of the resulting iteration in the H^1-norm. Our convergence analysis is based on a sequential strict local minimality inequality. We show abstractly how the order of minimality determines the convergence rate. Assuming that the initial guess lies above the exact solution and a one-sided Bregman growth condition holds, we establish sequential strict local minimality of order s ≥ 2 and derive sublinear convergence rates in the H^1-norm of the form O(k^−ζ), where ζ∈(1/2,1] depends on the choice of Legendre function. For the Shannon and Tsallis entropies, we show that these rates are sharp in a uniform worst-case sense. Under additional assumptions on the free boundary and the obstacle, we establish sequential strict local minimality of order s = 1 with the Shannon and Spence entropies. This yields linear convergence of the form O(ϱ^k) for some ϱ∈(0,1).
ER -