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Lagrangian Relaxation Score-based Generation for Mixed Integer linear Programming

Ruobing Wang, Xin Li, Yujie Fang, Mingzhong Wang

Abstract

Predict-and-search (PaS) methods have shown promise for accelerating mixed-integer linear programming (MILP) solving. However, existing approaches typically assume variable independence and rely on deterministic single-point predictions, which limits solution diversityand often necessitates extensive downstream search for high-quality solutions. In this paper, we propose \textbf{SRG}, a generative framework based on Lagrangian relaxation-guided stochastic differential equations (SDEs), with theoretical guarantees on solution quality. SRG leverages convolutional kernels to capture inter-variable dependencies while integrating Lagrangian relaxation to guide the sampling process toward feasible and near-optimal regions. Rather than producing a single estimate, SRG generates diverse, high-quality solution candidates that collectively define compact and effective trust-region subproblems for standard MILP solvers. Across multiple public benchmarks, SRG consistently outperforms existing machine learning baselines in solution quality. Moreover, SRG demonstrates strong zero-shot transferability: on unseen cross-scale/problem instances, it achieves competitive optimality with state-of-the-art exact solvers while significantly reducing computational overhead through faster search and superior solution quality.

Lagrangian Relaxation Score-based Generation for Mixed Integer linear Programming

Abstract

Predict-and-search (PaS) methods have shown promise for accelerating mixed-integer linear programming (MILP) solving. However, existing approaches typically assume variable independence and rely on deterministic single-point predictions, which limits solution diversityand often necessitates extensive downstream search for high-quality solutions. In this paper, we propose \textbf{SRG}, a generative framework based on Lagrangian relaxation-guided stochastic differential equations (SDEs), with theoretical guarantees on solution quality. SRG leverages convolutional kernels to capture inter-variable dependencies while integrating Lagrangian relaxation to guide the sampling process toward feasible and near-optimal regions. Rather than producing a single estimate, SRG generates diverse, high-quality solution candidates that collectively define compact and effective trust-region subproblems for standard MILP solvers. Across multiple public benchmarks, SRG consistently outperforms existing machine learning baselines in solution quality. Moreover, SRG demonstrates strong zero-shot transferability: on unseen cross-scale/problem instances, it achieves competitive optimality with state-of-the-art exact solvers while significantly reducing computational overhead through faster search and superior solution quality.
Paper Structure (34 sections, 11 theorems, 102 equations, 12 figures, 12 tables, 1 algorithm)

This paper contains 34 sections, 11 theorems, 102 equations, 12 figures, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Our Method
  5. Lagrangian Relaxation-guided SDEs
  6. Conditional Score Network
  7. Training and Sampling
  8. Experiments
  9. Settings
  10. Main Results
  11. Ablation Study and Sensitivity Analysis
  12. Generation Analysis
  13. Conclusion
  14. Proof of the main theorems.
  15. Details for our 2D Toy experiment.
  16. ...and 19 more sections

Key Result

Theorem 1

Consider the Lagrangian dual problem: Let $\{\lambda_{(k)}\}$ be the sequence of multipliers generated by the subgradient method: where $x_{(k)} \in \arg\min_{x \in X} \{ cx + (\lambda_{(k)})^\top (b - Ax) \}$, and the step sizes $\{\alpha_k\}$ satisfy: Then the sequence of dual values converges to the optimal dual value:

Figures (12)

  • Figure 1: Generation trajectory of our method on a 2D Linear Programming toy experiment (a special case of MILP). As the denoising process evolves, the learned score density (yellow heatmap) progressively concentrates toward the optimal solution (red star), effectively steering the samples (blue dots) from an initially scattered state into a tight cluster near the optimum. Additional experiment details are provided in Appx. \ref{['app:toyexperiments']}.
  • Figure 2: Best-so-far primal bound trajectories of all methods+SCIP/Gurobi on medium-scale MIS and CA MILP benchmarks. Results are averaged over 100 test instances.
  • Figure 3: Visualization of solution diversity on a medium-scale SC instance. Three rows correspond to three independent sampling runs, generating distinct candidates for the same problem without any post-processing.
  • Figure 4: Toy experiments with largrangian-guided score
  • Figure 5: Toy 2D LP instance with obj contour and each constraint
  • ...and 7 more figures

Theorems & Definitions (21)

  • Theorem 1: Subgradient Convergence shor1985minimization
  • Theorem 2: Optimization Equivalence
  • Theorem 3: Conditional Tighter Solution Region
  • Theorem 4: Lagrangian-Guided Reverse VP-SDE
  • proof
  • proof
  • proof
  • Lemma 1: spatial Space Isometry
  • proof
  • Lemma 2
  • ...and 11 more