Solving Integer Linear Programming with Parallel Tempering

2026-05-28

AI Summary

This work addresses key challenges in integer linear programming (ILP) solvers—limited generalization, reliance on external solvers, and poor efficiency in multimodal energy landscapes—by proposing a training-free, solver-free sampling-based optimization framework that directly explores the discrete feasible region. Leveraging the linear structure of ILP, the method designs a proposal distribution satisfying detailed balance and incorporates a dual tempering mechanism that jointly modulates temperature and penalty parameters to dynamically adjust constraint barriers while preserving the original objective function, thereby enhancing global exploration. Experiments demonstrate that the approach consistently outperforms SCIP across four benchmarks, matches or exceeds Gurobi’s performance on two tasks within 200 seconds, exhibits superior robustness on out-of-distribution instances, and competes effectively with classical solvers on MIPLIB 2017 without any parameter tuning.

Abstract

Integer Linear Programming (ILP) serves as a versatile framework for modeling a wide range of combinatorial optimization problems, typically addressed by sophisticated exact solvers or heuristics. While learning-based approaches have recently shown their effectiveness, they suffer from poor generalization to out-of-distribution instances and inherent dependence on external solvers. In this work, we propose a solver-free, sampling-based optimization framework for ILP that directly explores discrete feasible regions without training or external solvers. Exploiting the linear structure of ILP, we employ a Locally-Balanced Proposal to construct a transition kernel, thereby avoiding the gradient approximation. To overcome the highly multimodal nature of ILP energy landscapes, we integrate Parallel Tempering. In addition to standard temperature tempering, we introduce penalty tempering, which modulates constraint barriers while preserving the objective landscape over feasible solutions. Empirically, our method consistently outperforms SCIP across all four benchmarks, matches or exceeds Gurobi on two of four tasks within a 200-second budget, and is substantially more robust to distribution shift than learning-based methods. Furthermore, on MIPLIB 2017 instances, our framework remains competitive with classical solvers without any problem-specific tuning.

Problem

• Integer Linear Programming
• Combinatorial Optimization
• Multimodal Energy Landscapes
• Generalization
• Constraint Satisfaction

Innovation

• Parallel Tempering
• Locally-Balanced Proposal
• Penalty Tempering
• Solver-Free Optimization
• Integer Linear Programming