Optimal Boundary Control of Diffusion on Graphs via Linear Programming

2025-11-05

AI Summary

This work addresses the optimal boundary control problem for steady-state diffusion processes on geometric networks—i.e., steering interior fluxes via boundary node potential adjustments under physical constraints (sign consistency, flux capacity) and geometric fidelity. We propose a unified linear programming framework that integrates discrete diffusion laws, the network Laplacian operator, and Minkowski–Weyl decomposition, augmented with a gradient-derived flux constraint mechanism to ensure bounded feasible regions and existence of global minima. Theoretically, we prove solution existence under the absence of negative recession directions and establish numerical robustness via Hoffman-type bounds. The method applies to both full-rank and rank-deficient graph structures. Extensive validation on large-scale stadium and historic urban street networks demonstrates its effectiveness, scalability, and practical applicability.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Content-based information diffusion

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.

Problem

• Optimizing boundary controls for diffusion processes on geometric networks
• Enforcing physical constraints on boundary fluxes and potentials
• Solving large-scale network diffusion via linear programming framework

Innovation

• Linear programming framework for diffusion optimization
• Boundary potentials control interior fluxes via Laplacian
• Polyhedral feasible set with geometric network constraints