General Resolvents for Monotone Operators: Characterization and Extension
Bibliographic record
Abstract
Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the resolvents of monotone operators. For example, the proximal mappings in the sense of Moreau are precisely the resolvents of subdifferential operators. More general notions of "resolvent", "proximal mapping" and "firmly nonexpansive" have been studied. One important class, popularized chiefly by Alber and by Kohsaka and Takahashi, is based on the normalized duality mapping. Furthermore, Censor and Lent pioneered the use of the gradient of a well behaved convex functions in a Bregman-distance based framework. It is known that resolvents are firmly nonexpansive, but the converse has been an open problem for the latter framework. In this note, we build on the very recent characterization of maximal monotonicity due to Martinez-Legaz to provide a framework for studying resolvents in which firmly nonexpansive mappings are always resolvents. This framework includes classical resolvents, resolvents based on the normalized duality mapping, resolvents based on Bregman distances, and even resolvents based on (nonsymmetric) rotators. As a by-product of recent work on the proximal average, we obtain a constructive Kirszbraun-Valentine extension result for generalized firmly nonexpansive mappings. Several examples illustrate our results.
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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 | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.007 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".