Bregman-projected mirror methods for regularized stationary mean-field games
Abstract
We develop and analyze a Bregman-projected mirror iteration for low-order regularizations of stationary mean-field game (MFG) systems in their natural Banach space setting. For separable Hamiltonians of the form H(x,p,m)=H0(x,p)−g(m), with quadratic or super-quadratic Hamiltonian growth and linear or super-linear density couplings, we formulate a low-order γ̅-Laplacian regularization of the stationary MFG system as a variational inequality on Lβ̅(𝕋d)× W1,γ̅(𝕋d). To approximate solutions of this regularized variational inequality, we introduce a Bregman geometry matched to the mixed Lebesgue-Sobolev exponents of the problem and analyze the exact constrained one-step mirror iteration and a frozen two-step variant. Under quantitative monotonicity of the coupling, both exact iterations converge strongly to the unique solution of the regularized variational inequality for each fixed ε>0, under natural summability conditions on the step sizes. Along weakly convergent vanishing-regularization sequences, we also prove strong convergence in the full product space. Numerical experiments in one and two dimensions show first-order accuracy on manufactured solutions and weak dependence of outer iteration counts on the tested grids. At matched effective step sizes, the frozen two-step and one-step implementations agree up to inner-solve accuracy, while the latter requires fewer mirror solves.
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BibTeX
@misc{abdulaziz2026bregmanprojected,
title = {{Bregman-projected mirror methods for regularized stationary mean-field games}},
author = {Hussain Al Abdulaziz and Yuri Ashrafyan and Yeva Gevorgyan and Diogo Gomes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.19611},
eprint = {2606.19611},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.19611},
}
RIS
TY - UNPB
TI - Bregman-projected mirror methods for regularized stationary mean-field games
AU - Hussain Al Abdulaziz
AU - Yuri Ashrafyan
AU - Yeva Gevorgyan
AU - Diogo Gomes
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2606.19611
UR - https://arxiv.org/abs/2606.19611
AB - We develop and analyze a Bregman-projected mirror iteration for low-order regularizations of stationary mean-field game (MFG) systems in their natural Banach space setting. For separable Hamiltonians of the form H(x,p,m)=H_0(x,p)−g(m), with quadratic or super-quadratic Hamiltonian growth and linear or super-linear density couplings, we formulate a low-order γ̅-Laplacian regularization of the stationary MFG system as a variational inequality on L^β̅(𝕋^d)× W^1,γ̅(𝕋^d). To approximate solutions of this regularized variational inequality, we introduce a Bregman geometry matched to the mixed Lebesgue-Sobolev exponents of the problem and analyze the exact constrained one-step mirror iteration and a frozen two-step variant. Under quantitative monotonicity of the coupling, both exact iterations converge strongly to the unique solution of the regularized variational inequality for each fixed ε>0, under natural summability conditions on the step sizes. Along weakly convergent vanishing-regularization sequences, we also prove strong convergence in the full product space. Numerical experiments in one and two dimensions show first-order accuracy on manufactured solutions and weak dependence of outer iteration counts on the tested grids. At matched effective step sizes, the frozen two-step and one-step implementations agree up to inner-solve accuracy, while the latter requires fewer mirror solves.
ER -