Come Back to Lagrange. The<i>p</i>-Factor Analysis of Optimality Conditions
Bibliographic record
Abstract
We consider necessary optimality conditions for optimization problems with equality constraints given in the operator form as F(x) = 0, where F is an operator between Banach spaces. The article addresses the case when the Lagrange multiplier λ0 associated with the objective function might be equal to zero. If the equality constraints are not regular at some point in the sense that the Fréchet derivative of F at is not onto, then the point is a degenerate solution of the classical Lagrange system of optimality conditions ℒ(x, λ0, λ) = 0, where is a solution of the optimization problem and is a corresponding generalized Lagrange multiplier. We derive new conditions that guarantee that is a locally unique solution of the Lagrange system. We also introduce a modified Lagrange system and prove that is its regular locally unique solution. In addition, we propose new conditions that guarantee that the point is an isolated local minimizer of the optimization problem. The modified Lagrange system introduced in this article can be used as a basis for constructing numerical methods for solving degenerate optimization problems. Our results are based on the construction of p-regularity and are illustrated by examples.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.008 | 0.003 |
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".