Abstract
In this paper, we present an extension of Uzawa’s algorithm and apply it to build approximating sequences of mean field games systems. We prove that Uzawa’s iterations can be used in a more general situation than the one in it is usually used. We then present some numerical results of those iterations on discrete mean field games systems of optimal stopping, impulse control and continuous control.
Highlights
The second part of this paper is concerned with the application of this remark to build approximating sequences of solutions of Mean Field Games (MFG) systems
The study of a MFG requires to solve the so-called master equation, see [14, 24], but in the case when there is no common noise, the problem reduces to a system of Partial Differential Equations (PDE)
We have in this paper a similar interpretation in terms of variational inequalities for MFG systems and this approach is crucial to our study
Summary
This paper is concerned with the study of an extension of Uzawa’s algorithm. We show that the standard Uzawa’s algorithm can be used to find solutions of systems similar to the ones characterizing saddle points of lagrangians, even though there is not a proper langrangian associated with this system. The study of a MFG requires to solve the so-called master equation, see [14, 24], but in the case when there is no common noise, the problem reduces to a system of Partial Differential Equations (PDE). It is well known that in the so-called potential case, MFG systems can be interpreted as the optimality conditions for an optimal control problem. Uzawa’s algorithm is a natural method we can apply to such optimal control problems
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