Many specific problems ranging from theoretical probability to applications in statistical physics, combinatorial optimization and communications can be formulated as an optimal tuning of local parameters in large systems of interacting particles. Using the framework of stationary point processes in the Euclidean space, we pose it as a problem of an optimal stationary marking of a given stationary point process. The quality of a given marking is evaluated in terms of scores calculated in a covariant manner for all points as a function of the proposed marked configuration. In the absence of a total order of the configurations of scores, we identify intensity-optimality and local optimality as two natural ways for defining optimal stationary marking. We derive tightness and integrability conditions under which intensity-optimal markings exist and further stabilization conditions making them equivalent to locally optimal ones. We present examples motivating the proposed, general framework. Finally, we discuss various possible approaches leading to uniqueness results.

从理论概率到统计物理学中的应用，组合优化和通信等许多具体问题，都可以表述为在相互作用粒子的大型系统中局部参数的最佳调整。使用欧几里得空间中的固定点过程的框架，我们将其视为给定固定点过程的最佳固定标记的问题。给定标记的质量是根据所有标记的协方差方式计算的分数来评估的，该分数是所建议的标记配置的函数。在没有分数配置的总顺序的情况下，我们将强度最优和局部最优确定为定义最优静态标记的两种自然方式。我们推导出存在强度最佳标记的紧密性和可整合性条件，以及进一步的稳定化条件，使其等同于局部最优标记。我们提供了一些示例来激励提出的通用框架。最后，我们讨论了导致唯一性结果的各种可能方法。