Abstract

The paper presents a computational algorithm for solving problem formulated in terms of Mean Field Game theory with limited management resource. The Mean Field equilibrium leads to a coupled system of two parabolic partial differential equations: Fokker–Planck–Kolmogorov and Hamilton–Jacobi–Bellman ones with an additional constraint in the form of the inequality. The article is devoted to the discrete approximation of these equations and reformulating the discrete statement in the form of the saddle point problem with the condition of complementary slackness. An iterative algorithm is presented for solving the obtained discrete problem with justification of the convergence of its elements. The convergence of the algorithm as a whole is illustrated by a numerical example.

中文翻译：

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管理资源有限的平均场问题的数值解

摘要

本文提出了一种计算算法，用于解决根据平均场博弈理论制定的管理资源有限的问题。平均场平衡导致两个抛物线偏微分方程的耦合系统：Fokker-Planck-Kolmogorov 和 Hamilton-Jacobi-Bellman 方程，带有不等式形式的附加约束。本文致力于这些方程的离散逼近，并以具有互补松弛条件的鞍点问题的形式重新表述离散陈述。提出了一种迭代算法来解决所获得的离散问题，并证明其元素的收敛性。通过数值例子说明了算法整体的收敛性。