Hypergraph-based latent space optimization for end-to-end sustainable logistics network design

Designing optimal hypergraph structures for complex logistics networks remains challenging owing to non-differentiable combinatorial search spaces. Traditional methods fail to reconcile discrete topology selection with continuous physical flow constraints. This article proposes a fully differentiable pipeline for optimal hypergraph synthesis via geometric latent-space relaxation. By parameterizing discrete hypergraph structures through a continuous latent manifold and a hypergraph neural network decoder, the author transforms structural design into a gradient-based learning task. A bi-level optimization problem is formulated in which the inner loop solves convex flow equilibrium while the outer loop optimizes latent parameters. Crucially, exact analytical gradients are derived by applying the implicit function theorem to Karush–Kuhn–Tucker optimality conditions, enabling dual variables to act as shadow prices that direct structural growth towards capacity bottlenecks. The framework achieves 93% memory reduction compared to semidefinite programming while maintaining stability under extreme demand surges, providing a scalable, mathematically rigorous foundation for automated infrastructure design.