Facial reduction for the Shor SDP relaxation of QCQPs
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Bibliographic record
Abstract
We propose a special facial reduction algorithm (FRA) for the Shor semidefinite programming (SDP) relaxation of the quadratically constrained quadratic program (QCQP).Under the mild assumption, our special FRA only requires solving a linear programming problem instead of a semidefinite program.In particular, when applied to the binary quadratic program, the proposed special FRA needs fewer assumptions.From a computational perspective, this result improves the scalability and stability of the SDP approach for QCQP problems.In addition, we also discover a new class of semidefinite programs whose singularity degree can be computed easily.This new class complements the limited examples of singularity degrees in the literature.As a by-product, our special FRA can be used to upper bound the dimension of any set defined by quadratic inequalities.
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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 | 0.003 |
| 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