Constrained Mean Field Games: Analysis and Algorithms

本提案的目的是为具有控制和状态约束的微分纳什均衡问题产生的平均场博弈开发新的分析方法和求解算法。将控制约束和状态约束引入平均场对策中，产生了一类新的具有非光滑算子和非线性耦合的动态无限维混合互补问题。由于包含控制和状态约束，新的解析和数值解决范例将为这个快速增长的应用数学领域开发。在n参与人非合作微分纳什均衡问题中，平均场博弈以一种自然的方式出现，即让智能体的数量增长到无穷大。在导出非合作对策的适当一阶最优性条件后，出现在极限的有效方程构成了平均场对策。一个重要的方面是假设N个战略代理在其目标、约束和动态方面在统计上是同质的。这使得平均场博弈成为研究竞争实体的复杂动态系统的理想方法，随着人口规模的增长，这些实体或多或少都是同质的，例如在宏观经济学、生物学和大型网络中。从数学的角度来看，平均场博弈可以看作是概率测度流空间上的不动点迭代，表示状态密度随时间的演变。从初始人口分布开始，该流由连续性方程的解决定，其驱动场与代表代理的一系列最优控制问题相关联，代表代理反过来对该措施流做出反应。从更广泛的意义上说，对于任何给定的平均场博弈（MFG），有四个主要问题需要解决。近似均衡：MFG的解是否与原问题相关？存在性，唯一性：制造商是否拥有（唯一的）解决方案？收敛问题：纳什博弈是否真的收敛于MFG？计算：我们能解决实际应用中的约束mfg吗？在这些类别的背景下，我们将考虑几种一般类型的约束mfg，它们具有对许多应用很重要的一般二次目标函数，具有鲁棒失拟项的函数，以表示智能体对平均场相互作用的异常值的敏感性，以及稀疏控制动作。我们允许以圆锥、时间相关双边约束和一般多面体约束形式的几种控制和状态约束。对于个体动力学，我们将考虑应用中出现的确定性和随机线性动力学以及确定性非线性动力学。

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.