Generalized Hukuhara weak subdifferential and its application on identifying optimality conditions for nonsmooth interval-valued functions
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Bibliographic record
Abstract
In this paper, we introduce the idea of gH-weak subdifferential for interval-valued functions (IVFs) and show how to calculate gH-weak subgradients.It is observed that a nonempty gH-weak subdifferential set is convex and closed.In characterizing the class of functions for which the gH-weak subdifferential set is nonempty, it is identified that this class is the collection of gH-lower Lipschitz IVFs.In checking the validity of the sum rule of gH-weak subdifferential for a pair of IVFs, a counterexample is obtained, which reflects that the sum rule does not hold.However, under a mild restriction on one of the IVFs, one-sided inclusion for the sum rule holds.As applications, we employ gH-weak subdifferential to provide a few optimality conditions for nonsmooth IVFs.Further, a necessary optimality condition for interval optimization problems with a difference of two nonsmooth IVFs as the objective is established.Next, a necessary and sufficient condition via augmented normal cone and gH-weak subdifferential of IVFs for finding weak efficient points is presented.Lastly, in investigating a 'sup-relation' between gH-direction derivative and gH-weak subgradients, we approximately compute gH-weak subgradient at each iterative step.In the sequel, we propose W -gH-weak subgradient method to identify a weak efficient solution of an unconstrained nonsmooth IOP.We apply the proposed method to solve an interval optimization problem by taking a test example.We present a convergence analysis of the proposed method for constant and diminishing step sizes.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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