Dual-Optimal Inequalities for Stabilized Column Generation
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Bibliographic record
Abstract
Column generation is one of the most successful approaches for solving large-scale linear programming problems. However, degeneracy difficulties and long-tail effects are known to occur as problems become larger. In recent years, several stabilization techniques of the dual variables have proven to be effective. We study the use of two types of dual-optimal inequalities to accelerate and stabilize the whole convergence process. Added to the dual formulation, these constraints are satisfied by all or a subset of the dual-optimal solutions. Therefore, the optimal objective function value of the augmented dual problem is identical to the original one. Adding constraints to the dual problem leads to adding columns to the primal problem, and feasibility of the solution may be lost. We propose two methods for recovering primal feasibility and optimality, depending on the type of inequalities that are used. Our computational experiments on the binary and the classical cutting-stock problems, and more specifically on the so-called triplet instances, show that the use of relevant dual information has a tremendous effect on the reduction of the number of column generation iterations.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it