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

Alisina Bayati, Dhananjay Tiwari, Srinivasa Salapaka

TL;DR

This work tackles NP-hard capacitated facility location problems by marrying Deterministic Annealing with a control-theoretic CLF-CBF framework to enforce feasibility while guiding the solution toward KKT points. By relaxing discrete assignments into soft distributions and recasting the inner optimization as a control problem, it introduces a QP-based policy that enforces descent of a CLF-like objective and forward-invariance of inequality and equality constraints via CBFs, with annealing across $\beta$ for progressively sharper solutions. The method demonstrates substantial computational efficiency gains and scalability to large instances (e.g., $N=2000$, $M=10$) while maintaining competitive objective costs compared to traditional solvers, and it supports dynamic extensions where demand evolves over time. These results underscore the potential of combining control-theoretic safety guarantees with entropy-based optimization for large-scale, constrained resource allocation tasks, enabling robust, real-time or near-real-time decision making in complex networks.

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.

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

TL;DR

This work tackles NP-hard capacitated facility location problems by marrying Deterministic Annealing with a control-theoretic CLF-CBF framework to enforce feasibility while guiding the solution toward KKT points. By relaxing discrete assignments into soft distributions and recasting the inner optimization as a control problem, it introduces a QP-based policy that enforces descent of a CLF-like objective and forward-invariance of inequality and equality constraints via CBFs, with annealing across for progressively sharper solutions. The method demonstrates substantial computational efficiency gains and scalability to large instances (e.g., , ) while maintaining competitive objective costs compared to traditional solvers, and it supports dynamic extensions where demand evolves over time. These results underscore the potential of combining control-theoretic safety guarantees with entropy-based optimization for large-scale, constrained resource allocation tasks, enabling robust, real-time or near-real-time decision making in complex networks.

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.

Paper Structure

This paper contains 17 sections, 3 theorems, 19 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Maximum Entropy Principle (MEP) based solution approach
  4. Unconstrained Facility Location Problem
  5. Constrained Facility Location Problem
  6. A Control-Theoretic Viewpoint on Optimization
  7. Our Approach
  8. SGF Aproach
  9. Instantiating our CBF-based Approach for $\textrm{P}_{\textrm{constr}}(\beta)$
  10. SIMULATIONS AND RESULTS
  11. Constraint handling, speed and cost:
  12. Convergence rates at fixed $\beta$:
  13. Solving a large scale problem:
  14. Conclusion and Future works
  15. APPENDIX
  16. ...and 2 more sections

Key Result

Theorem 1

Under the assumptions assump:constraint_gradients and assump:coercivity, consider the feedback control $u^*(z)$ defined as the solution to the following quadratic program (QP): where $\gamma$ is a class $\mathcal{K}$ function, each $\alpha_j$ is an extended class $\mathcal{K}_\infty$ function and $q > 0$ is a constant. Then, the following properties hold:

Figures (2)

  • Figure 1: The figure shows a capacitated FLP with 400 demand points in 4 clusters, solved using the four methods. Final resource utilization is shown to the right of each subplot. All the approach maintain feasibility except the DA-based penalty method. Runtimes (in sec): $\{46, 210, 1600, 60\}$, Final costs: $\{46, 99, 60, 33\}$ units.
  • Figure 2: Capacitated FLP solution using our CBF-based approach for $N=1000, M=10.$ The cluster split of users: $\left[0.11, 0.07, 0.11, 0.09, 0.13, 0.14, 0.02, 0.11, 0.14, 0.08\right]$ and facility (R) utilization (U) constraints are shown at the bottom right. The figure also shows splitting of facilities into distinct clusters as $\beta \in \left[10^{-3},100\right]$ is increased during annealing.

Theorems & Definitions (11)

  • Remark 1
  • Definition 1
  • Definition 2
  • Theorem 1
  • proof : Theorem \ref{['Thrm:main_theorem']}
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1: Farkas' Lemma, Variant (iii)
  • ...and 1 more