Primal-Dual Symmetric Intrinsic Methods for Finding Antiderivatives of Cyclically Monotone Operators
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Bibliographic record
Abstract
A fundamental result due to Rockafellar states that every cyclically monotone operator A admits an antiderivative f in the sense that the graph of A is contained in the graph of the subdifferential operator $\partial f$. Given a method $\mathfrak{m}$ that assigns every finite cyclically monotone operator A some antiderivative $\mathfrak{m}_A$, we say that the method is primal-dual symmetric if $\mathfrak{m}$ applied to the inverse of A produces the Fenchel conjugate of $\mathfrak{m}_A$. Rockafellar's antiderivatives do not possess this property. Utilizing Fitzpatrick functions and the proximal average, we present novel primal-dual symmetric intrinsic methods. The antiderivatives produced by these methods provide a solution to a problem posed by Rockafellar in 2005. The results leading to this solution are illustrated by various examples.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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