Simplified Newton method for solving large-scale stochastic nonlinear matrix equations in mean-field social control
Zihang Tian, Hiroaki Mukaidani · Journal of Computational and Applied Mathematics · 2025
This paper investigates a numerical framework for solving incentive Stackelberg games in mean-field stochastic systems characterized by large numbers of followers. A well-known challenge in such systems is the computational bottleneck that arises as the follower count N approaches infinity, rendering implementation infeasible owing to exceeding physical limits. To address this issue, we propose a novel numerical algorithm that generates a decentralized Pareto strategy independent of the number of followers. Specifically, we developed a low-dimensional algorithm based on a simplified Newton’s method. This approach effectively handles the state dimension R n × n of each follower by partitioning the very large-scale stochastic nonlinear matrix equations (SNMEs), which would otherwise require extensive computations in R ( N + 1 ) n × ( N + 1 ) n dimensions. We provide, for the first time, proof demonstrating the linear convergence of this method. Additionally, a Lyapunov iterative method is explored as an alternative to circumvent complex mathematical derivations. The proposed methods significantly reduce computational complexity by enabling low-dimensional computations for each follower, contrasting with the high-dimensional computations required by previous approaches. To prevent divergence of the norm of the solution, even with a small number of followers, we introduce a new condition for the coefficient matrices of the original mean-field stochastic system and develop an incentive strategy to ensure a finite norm. Finally, we validate the effectiveness and reliability of our algorithm by solving a large-scale set of SNMEs and confirming the linear convergence of the simplified Newton’s method.