Learning-based Directed Graph Abstraction of Combinatorial Spaces for Order-Preserving Search in Mixed-Combinatorial Nonlinear Optimization

2026-05-31

AI Summary

Traditional combinatorial space representations—such as integer or binary encodings—introduce spurious relationships, dimensional inflation, and extraneous constraints in mixed-combinatorial nonlinear optimization, thereby degrading search efficiency. This work proposes a direction-aware directed graph abstraction that leverages an Edge Field Graph Network (EFGN) to map an undirected fully connected combinatorial graph into a structured directed improvement-direction graph. This graph is embedded within the optimization framework as a recommendation system, enabling search exclusively over continuous variables while dynamically retrieving optimal combinatorial configurations. The approach achieves, for the first time, a scalable and interpretable structured modeling of combinatorial spaces. Evaluated on three nonlinear benchmark problems, it significantly outperforms index-based combinatorial baselines, yielding superior average solutions and enhanced robustness.

Abstract

Mixed-combinatorial nonlinear programming (MCNLP) problems arise in many engineering design and planning applications, e.g., due to categorical, component, and geometric design choices, as well as joint task and motion planning. Traditional representations of combinatorial spaces, such as integer or binary encoding, often introduce spurious relations, increase dimensionality, and require additional compatibility constraints. Instead, this paper draws on recent developments in robot planning and vehicle/network routing domains that aim to learn search heuristics over combinatorial spaces using graph neural networks (GNNs). More specifically, this paper presents a first-of-its-kind structured abstraction of the combinatorial space by learning a mapping from an undirected fully connected graph of combinations to a directed graph indicating improvement directions using an Edge Field Graph Network (EFGN). To demonstrate the utility of this new way of abstracting the combinatorial space in solving MCNLPs, we adopt a recent optimization framework that purely searches over the non-combinatorial (e.g., continuous) variables and retrieves the best-suited combination for each candidate design by using the abstraction model, akin to a recommender system. The presented direction-aware abstraction model provides a potentially more scalable and interpretable retrieval of combinations compared to the original recommendation system in that framework. For evaluation, the proposed method is integrated with a well-known particle swarm optimization and genetic algorithm solvers on three benchmark nonlinear problems with varying numbers of combinations and variables. Compared to baseline solvers using indexified combinations, the GNN-based recommender consistently achieves better mean optimum values and robustness across multiple runs.

Problem

• Mixed-combinatorial nonlinear programming
• Combinatorial space representation
• Spurious relations
• Compatibility constraints
• Scalability

Innovation

• Graph Neural Networks
• Combinatorial Abstraction
• Directed Graph Learning
• Mixed-Combinatorial Optimization
• Edge Field Graph Network