Analysis of Bregman Proximal Point for the Obstacle Problem
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 $s \geq 2$ and derive sublinear convergence rates in the $H^1$-norm of the form $O(k^{-ζ})$, where $ζ\in(1/2,1]$ 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 $O(\varrho^k)$ for some $\varrho\in(0,1)$.