Simulation-based problems involving mixed-variable inputs frequently feature domains that are hierarchical, conditional, heterogeneous, or tree-structured. These characteristics pose challenges for data representation, modeling, and optimization. This paper reviews extensive literature on these structured input spaces and proposes a unified framework that generalizes existing approaches. In this framework, input variables may be continuous, integer, or categorical. A variable is described as meta if its value governs the presence of other decreed variables, enabling the modeling of conditional and hierarchical structures. We further introduce the concept of partially-decreed variables, whose activation depends on contextual conditions. To capture these inter-variable hierarchical relationships, we introduce design space graphs, combining principles from feature modeling and graph theory. This allows the definition of general hierarchical domains suitable for describing complex system architectures. Our framework defines hierarchical distances and kernels to enable surrogate modeling and optimization on hierarchical domains. We demonstrate its effectiveness on complex system design problems, including a neural network and a green-aircraft case study. Our methods are available in the open-source Surrogate Modeling Toolbox (SMT 2.0).

翻译：基于仿真的混合变量输入问题常涉及具有层次化、条件性、异质性或树状结构特征的领域。这些特性给数据表示、建模与优化带来了挑战。本文系统综述了关于此类结构化输入空间的大量文献，并提出一个能泛化现有方法的统一框架。在该框架中，输入变量可以是连续型、整数型或类别型。若某变量的取值决定了其他衍生变量的存在性，则将其定义为元变量，从而实现对条件结构与层次结构的建模。我们进一步引入部分衍生变量的概念，其激活状态取决于上下文条件。为捕捉这些变量间的层次关系，我们提出设计空间图的概念，融合了特征建模与图论原理。这使得定义适用于描述复杂系统架构的通用层次化领域成为可能。本框架定义了层次距离与核函数，以支持在层次化领域上进行代理建模与优化。我们在复杂系统设计问题（包括神经网络与绿色飞行器案例研究）中验证了该框架的有效性。相关方法已在开源代理建模工具箱（SMT 2.0）中实现。