Optimal Boundary Control of Diffusion on Graphs via Linear Programming
Harbir Antil, Rainald Löhner, Felipe Pérez
TL;DR
This work develops a linear-programming framework for steady-state diffusion on weighted, oriented graphs, treating boundary potentials as controls and deriving an affine control-to-state mapping that yields a finite-dimensional, convex LP with a polyhedral feasible set. Existence (and boundedness under practical conditions) of a global minimizer is established via recession-cone analysis and Minkowski–Weyl decomposition, connecting to classic convex-analytic results. The method is demonstrated on two large networks—a stadium concourse and a complex urban street network—showing conservation and sign-correct fluxes that align with geometric and physical constraints, and highlighting potential for fast re-optimization in digital-twin contexts. The approach offers a principled, scalable tool for network-based diffusion control with applications in crowd dynamics, transport, and multi-scale PDE-constrained optimization.
Abstract
We propose a linear programming (LP) framework for steady-state diffusion and flux optimization on geometric networks. The state variable satisfies a discrete diffusion law on a weighted, oriented graph, where conductances are scaled by edge lengths to preserve geometric fidelity. Boundary potentials act as controls that drive interior fluxes according to a linear network Laplacian. The optimization problem enforces physically meaningful sign and flux-cap constraints at all boundary edges, derived directly from a gradient bound. This yields a finite-dimensional LP whose feasible set is polyhedral, and whose boundedness and solvability follow from simple geometric or algebraic conditions on the network data. We prove that under the absence of negative recession directions--automatically satisfied in the presence of finite box bounds, flux caps, or sign restrictions--the LP admits a global minimizer. Several sufficient conditions guaranteeing boundedness of the feasible region are identified, covering both full-rank and rank-deficient flux maps. The analysis connects classical results such as the Minkowski--Weyl decomposition, Hoffman's bound, and the fundamental theorem of linear programming with modern network-based diffusion modeling. Two large-scale examples illustrate the framework: (i) A typical large stadium in a major modern city, which forms a single connected component with relatively uniform corridor widths, and a (ii) A complex street network emanating from a large, historical city center, which forms a multi-component system.