Solving Algorithm NCL’s Subproblems: The Need for Interior Methods
Bibliographic record
Abstract
Abstract Algorithm NCL was devised to solve a class of large nonlinearly constrained optimization problems whose constraints do not satisfy LICQ at a solution. It is mathematically equivalent to the augmented Lagrangian algorithm LANCELOT, which solves a short sequence of bound-constrained subproblems $$\text {BC}_k$$ BC k and has no LICQ difficulties. NCL’s equivalent subproblems $$\text {NC}_k$$ NC k are much bigger and must be solved by a nonlinear interior method (needing first and second derivatives). We study the KKT-type systems arising within nonlinear interior methods when they are applied to the $$\text {NC}_k$$ NC k subproblems. We find that the KKT systems can sometimes be reduced to smaller SQD systems (symmetric quasi-definite) and sometimes to even smaller SPD systems (symmetric positive definite). The smaller systems have proved suitable for GPU implementation within the interior solver MadNLP when it is used by MadNCL to implement Algorithm NCL.
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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 | 0.012 |
| 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 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.011 | 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".