A Primal-Dual Algorithm for Solving Polyhedral Conic Systems with a Finite-Precision Machine
Bibliographic record
Abstract
We describe a primal-dual interior-point algorithm that determines which one of two alternative systems, [ Ax = 0, \quad x \geq 0, \] and \[ A\transp y \leq 0, \] is strictly feasible, provided that this pair of systems is well-posed. Furthermore, when the second system is strictly feasible, the algorithm returns a strict solution y; when the first system is strictly feasible, the algorithm returns a strict forward-approximate solution x. Here $A \in {\mathbb R}^{m\times n}$ is given. Our algorithm works with finite-precision arithmetic. The amount of precision required is adjusted as the algorithm progresses and remains bounded by a measure of well-posedness C(A) of the pair of systems of constraints. The algorithm halts in at most ${\cal O}((m+n)^{1/2}(\log(m+n)+\log(C(A))+|\log\gamma|))$ interior-point iterations, where $\gamma\in(0,1)$ is a parameter specifying the desired degree of accuracy of the forward-approximate solution for the first system. If the feasible system is the second one, the term $|\log\gamma|$ in the bound on the number of iterations can be dropped.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.002 |
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".