Hierarchical Modeling and Architecture Optimization: Review and Unified Framework

2025-06-27

AI Summary

Simulation optimization involving mixed-variable design spaces—comprising continuous, integer, and categorical variables—with hierarchical dependencies, conditional activation, and tree-structured relationships poses significant modeling and search challenges. Method: We propose a unified modeling paradigm that introduces *meta-variables* and *partially ordered conditional variables*, formalized via a *design space graph* to explicitly encode hierarchical and conditional variable relationships. Integrating graph-theoretic representations with feature-based modeling, we design a hierarchical kernel function and a non-Euclidean distance metric tailored to non-flat, structured domains, embedded within a surrogate-modeling framework compatible with Bayesian optimization. Contribution/Results: This is the first approach to enable unified modeling and optimization over complex conditional design spaces. Implemented in the open-source SMT 2.0 toolkit, it demonstrates substantial improvements in both modeling fidelity and search efficiency, as validated on green aircraft architecture optimization.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

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. The framework supports the use of surrogate models over such domains and integrates hierarchical kernels and distances for efficient modeling and optimization. The proposed methods are implemented in the open-source Surrogate Modeling Toolbox (SMT 2.0), and their capabilities are demonstrated through applications in Bayesian optimization for complex system design, including a case study in green aircraft architecture.

Problem

• Handling hierarchical mixed-variable inputs in simulations
• Modeling conditional and tree-structured data relationships
• Optimizing complex system architectures with surrogate models

Innovation

• Unified framework for hierarchical mixed-variable modeling
• Design space graphs for variable relationships
• Hierarchical kernels in surrogate modeling optimization