A Convergent Hierarchy of Certificates for Constrained Signomial Positivity
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Optimization is at the heart of many engineering problems. Many optimization problems, however, are computationally intractable. One approach to tackle such intractability is to find a tractable problem whose solution, when found, approximates that of the original problem. Specifically, convex optimization problems are often efficiently solvable, and finding a convex formulation that approximates a nonconvex problem, known as convex \nrelaxation, is an effective approach. \n \nThis work concerns a particular class of optimization problem, namely constrained signomial optimization. Based on the idea that optimization of a function is equivalent to verifying its positivity, we first study a certificate of signomial positivity over a constrained set, which finds a decomposition of the signomial into sum of parts that are verifiably positive via convex constraints. However, the certificate only provides a sufficient condition for positivity. The main contribution of the work is to show that by multiplying additionally more complex functions, larger subset of signomials that are positive over a \ncompact convex set, and eventually all, may be certified by the above method. The result is analogous to classic Positivstellensatz results from algebraic geometry which certifies polynomial positivity by finding its representation with sum of square polynomials. \n \nThe result provides a convergent hierarchy of certificate for signomial positivity over a constrained set that is increasingly more complete. The hierarchy of certificate in turn gives a convex relaxation algorithm that computes the lower bounds of constrained signomial optimization problems that are increasingly tighter at the cost of additional computational complexity. At some finite level of the hierarchy, we obtain the optimal solution.
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.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it