We present the concept of a Generalized Feedback Nash Equilibrium (GFNE) in dynamic games, extending the Feedback Nash Equilibrium concept to games in which players are subject to state and input constraints. We formalize necessary and sufficient conditions for (local) GFNE solutions at the trajectory level, which enable the development of efficient numerical methods for their computation. Specifically, we propose a Newton-style method for finding game trajectories which satisfy necessary conditions for an equilibrium, which can then be checked against sufficiency conditions. We show that the evaluation of the necessary conditions in general requires computing a series of nested, implicitly-defined derivatives, which quickly becomes intractable. To this end, we introduce an approximation to the necessary conditions which is amenable to efficient evaluation, and in turn, computation of solutions. We term the solutions to the approximate necessary conditions Generalized Feedback Quasi-Nash Equilibria (GFQNE), and we introduce numerical methods for their computation. In particular, we develop a Sequential Linear-Quadratic Game approach, in which a LQ local approximation of the game is solved at each iteration. The development of this method relies on the ability to compute a GFNE to inequality- and equality-constrained LQ games, and therefore specific methods for the solution of these special cases are developed in detail. We demonstrate the effectiveness of the proposed solution approach on a dynamic game arising in an autonomous driving application.

翻译：本文提出了动态博弈中广义反馈纳什均衡（GFNE）的概念，将反馈纳什均衡扩展至玩家受状态与输入约束的博弈场景。我们在轨迹层面形式化了（局部）GFNE解的充分必要条件，从而为其高效数值求解方法的开发奠定基础。具体而言，我们提出了一种牛顿型方法，用于寻找满足均衡必要条件的博弈轨迹，并可进一步通过充分条件进行验证。研究表明，一般情形下必要条件的评估需要计算一系列嵌套隐式导数，这会迅速变得难以处理。为此，我们引入了一种便于高效评估的必要条件近似形式，并在此基础上实现解的求解。我们将满足近似必要条件的解称为广义反馈拟纳什均衡（GFQNE），并提出了相应的数值求解方法。特别地，我们开发了序列线性-二次博弈方法，每次迭代中求解博弈的局部线性-二次近似。该方法的建立依赖于对含不等式与等式约束的线性-二次博弈的GFNE计算能力，因此我们详细推导了这些特例的求解方法。最终，通过一个自动驾驶场景中的动态博弈案例，验证了所提求解方法的有效性。