A Control Barrier Function Approach to Constrained Resource Allocation Problems in a Maximum Entropy Principle Framework

Abstract: This paper presents a novel approach to solve capacitated facility location problems (FLP) that encompass various resource allocation problems. FLPs are a class of NP-hard combinatorial optimization problems, involving optimal placement and assignment of a small number of facilities over a large number of demand points, with each facility subject to upper and lower bounds on its resource utilization (e.g., the number of demand points it can serve). To address the challenges posed by inequality constraints and the combinatorial nature of the solution space, we reformulate the problem as a dynamic control design problem, enabling structured constraint handling and enhanced solution efficiency. Our method integrates a Control Barrier Function (CBF) and Control Lyapunov Function (CLF)-based framework with a maximum-entropy principle-based framework to ensure feasibility, optimality, and improved exploration of solutions. Numerical experiments demonstrate that this approach significantly enhances computational efficiency, yielding better solutions and showing negligible growth in computation time with problem size as compared to existing solvers. These results highlight the potential of control-theoretic and entropy-based methods for large-scale facility location problems.