Constrained Mean Field Games: Analysis and Algorithms

• Applicant: Professor Dr. Michael Hintermüller, since 9/2022
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 423610162

The purpose of this proposal is to develop new analytical approaches and solution algorithms for mean-field games arising from differential Nash equilibrium problems with control and state constraints. The incorporation of control and state constraints into mean field games leads to new classes of dynamic infinite-dimensional mixed complementarity problems with nonsmooth operators and nonlinear couplings. Due to the inclusion of control and state constraints, novel analytical and numerical solution paradigms will be developed for this rapidly growing area of applied mathematics. Mean field games arise in a natural way by letting the number of agents grow to infinity in N-player non-cooperative differential Nash equilibrium problems. After deriving appropriate first-order optimality conditions for the non-cooperative game, the effective equations appearing in the limit constitute a mean field game. One important aspect is the assumption that the N strategic agents are statistically homogeneous in their objectives, constraints, and dynamics. This makes mean field games ideal for investigating complex dynamical systems of competing entities, who appear more or less homogeneous as the size of the population grows, e.g. in macroeconomics, biology, and large networks. From a mathematical perspective, mean field games can be viewed as a fixed point iteration on a space of flows of probability measures that represent the evolution of the density of the states in time. Starting from an initial population distribution, this flow is determined by the solution of a continuity equation, whose driving field is linked to a family of optimal control problems of a representative agent, who in turn is reacting to this flow of measures. In a broader sense, there are four main issues to be addressed for any given mean field game (MFG). Approximate Equilibria: Does the solution of the MFG relate to the original problem? Existence, Uniqueness: Does the MFG possess a (unique) solution? Convergence Problem: Does the Nash game actually converge to the MFG? Computation: Can we solve constrained MFGs for practical applications? Within the context of these categories, we will consider several general classes of constrained MFGs with general quadratic objective functionals important for many applications, functionals with robust misfit terms to represent the agents' sensitivity to outliers to the mean field interaction, and sparse control actions. We allow several categories of both control and state constraints in the form of conic, time-dependent bilateral constraints, and general polyhedral constraints. For the individual dynamics, we will consider both deterministic and stochastic linear dynamics and deterministic nonlinear dynamics arising in applications.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1962: Non-smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization
• Ehemaliger Antragsteller: Professor Dr. Thomas Michael Surowiec, until 9/2022