Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs
Bibliographic record
Abstract
Abstract We consider both facial reduction, FR, and symmetry reduction, SR, techniques for semidefinite programming, SDP. We show that the two together fit surprisingly well in an alternating direction method of multipliers, ADMM, approach. In fact, this approach allows for simply adding on nonnegativity constraints, and solving the doubly nonnegative, DNN , relaxation of many classes of hard combinatorial problems. We also show that the singularity degree remains the same after SR, and that the DNN relaxations considered here have singularity degree one, that is reduced to zero after FR. The combination of FR and SR leads to a significant improvement in both numerical stability and running time for both the ADMM and interior point approaches. We test our method on various DNN relaxations of hard combinatorial problems including quadratic assignment problems with sizes of more than $$n=500$$ n = 500 . This translates to a semidefinite constraint of order 250, 000 and $$625\times 10^8$$ 625 × 10 8 nonnegative constrained variables, before applying the reduction techniques.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".