Learning Valid Dual Bounds in Constraint Programming: Boosted Lagrangian Decomposition with Self-Supervised Learning
Bibliographic record
Abstract
Lagrangian decomposition (LD) is a relaxation method that provides a dual bound for constrained optimization problems by decomposing them into more manageable sub-problems. This bound can be used in branch-and-bound algorithms to prune the search space effectively.In brief, a vector of Lagrangian multipliers is associated with each sub-problem, and an iterative procedure (e.g., a sub-gradient optimization) adjusts these multipliers to find the tightest bound. Initially applied to integer programming, Lagrangian decomposition also had success in constraint programming due to its versatility and the fact that global constraints provide natural sub-problems. However, the non-linear and combinatorial nature of sub-problems in constraint programming makes it computationally intensive to optimize the Lagrangian multipliers with sub-gradient methods at each node of the tree search. This currently limits the practicality of LD as a general bounding mechanism for constraint programming. To address this challenge, we propose a self-supervised learning approach that leverages neural networks to generate multipliers directly, yielding tight bounds. This approach significantly reduces the number of sub-gradient optimization steps required, enhancing the pruning efficiency and reducing the execution time of constraint programming solvers. This contribution is one of the few that leverage learning to enhance bounding mechanisms on the dual side, a critical element in the design of combinatorial solvers. This work presents a generic method for learning valid dual bounds in constraint programming. We validate our approach on two challenging combinatorial problems: The multi-dimensional knapsack problem and the shift scheduling problem. The results show that our approach can solve more instances than the standard application of LD to constraint programming, reduce execution time by more than half, and has promising generalization ability through fine-tuning.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".