Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs
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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 $$n=500$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>500</mml:mn> </mml:mrow> </mml:math> . This translates to a semidefinite constraint of order 250, 000 and $$625\times 10^8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>625</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>8</mml:mn> </mml:msup> </mml:mrow> </mml:math> nonnegative constrained variables, before applying the reduction techniques.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 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.
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