Wegstein's Method for Calculating the Global Extremum
Bibliographic record
Abstract
This study discusses an economical and efficient method for calculating the global optimum of a function of many variables. The proposed algorithm can be attributed to methods based on auxiliary functions. The auxiliary function itself is obtained by converting the objective function using the Lebesgue integral and is a function of one variable. In a previously published paper by one of the authors of this article, this auxiliary function was used to calculate the global minimum of smooth multiextremal functions on convex closed sets. In the same article, an algorithm was proposed for dividing a segment into half to find a global minimum. And in this paper we consider the problem of finding the global minimum of continuous functions defined on bounded closed subsets of an n-dimensional Euclidean space. In addition, curious properties of the auxiliary function are established that are valid for any continuous objective function. For example, its non-negativity, positive homogeneity of some order, uniform continuity, differentiability and strict convexity are proved, and higher-order derivatives are calculated. The optimality criterion is established. The essence of this optimality criterion is that the value of a variable at which the auxiliary function and its derivatives are equal to zero up to a certain order turns out to be equal to the global minimum of the objective function. It follows from this optimality criterion that to calculate the global minimum of the objective function, it is sufficient to find the zero of the auxiliary function or its derivative up to the m-th order. Therefore, Wegstein's algorithm was used as a way to find the root of an equation with one unknown. In addition, the advantage of the Wegstein’s method is that it always converges. And in this situation, it turned out to be more efficient, despite its slow convergence, since it requires almost half the number of calculations of the values of the auxiliary function and that halves the need for numerical calculations of multiple integrals with a large number of variables.
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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".