Implementing a smooth exact penalty function for equality-constrained nonlinear optimization
Bibliographic record
Abstract
We develop a general equality-constrained nonlinear optimization algorithm based on a smooth penalty function proposed by Fletcher (1970). Although it was historically considered to be computationally prohibitive in practice, we demonstrate that the computational kernels required are no more expensive than other widely accepted methods for nonlinear optimization. The main kernel required to evaluate the penalty function and its derivatives is solving a structured linear system. We show how to solve this system efficiently by storing a single factorization each iteration when the matrices are available explicitly. We further show how to adapt the penalty function to the class of factorization-free algorithms by solving the linear system iteratively. The penalty function therefore has promise when the linear system can be solved efficiently, e.g., for PDE-constrained optimization problems where efficient preconditioners exist. We discuss extensions including handling simple constraints explicitly, regularizing the penalty function, and inexact evaluation of the penalty function and its gradients. We demonstrate the merits of the approach and its various features on some nonlinear programs from a standard test set, and some PDE-constrained optimization problems.
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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 | 0.001 |
| 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) | 0.003 | 0.001 |
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".