A notion of total dual integrality for convex, semidefinite, and extended formulations
Bibliographic record
Abstract
Total dual integrality is a powerful and unifying concept in polyhedral combinatorics and integer programming that enables the refinement of geometric min-max relations given by linear programming strong duality into combinatorial min-max theorems. The definition of a linear inequality system being totally dual integral (TDI) revolves around the existence of optimal dual solutions that are integral and thus naturally applies to a host of combinatorial optimization problems that are cast as integer programs whose linear program (LP) relaxations have the TDIness property. However, when combinatorial problems are formulated using more general convex relaxations, such as semidefinite programs (SDPs), it is not at all clear what an appropriate notion of integrality in the dual program is, thus inhibiting the generalization of the theory to more general forms of structured convex optimization. (In fact, we argue that the rank-one constraint usually added to SDP relaxations is not adequate in the dual SDP.) In this paper, we propose a notion of total dual integrality for SDPs that generalizes the notion for LPs, by relying on an “integrality constraint" for SDPs that is primal-dual symmetric. A key ingredient for the theory is a generalization to compact convex sets of a result of Hoffman for polytopes, fundamental for generalizing the polyhedral notion of total dual integrality introduced by Edmonds and Giles. We study the corresponding theory applied to SDP formulations for stable sets in graphs using the Lovász theta function and show that total dual integrality in this case corresponds to the underlying graph being perfect. We also relate dual integrality of an SDP formulation for the maximum cut problem to bipartite graphs. Total dual integrality for extended formulations naturally comes into play in this context.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.005 | 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".