The Discrete Moment Problem with Nonconvex Shape Constraints
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Bibliographic record
Abstract
The discrete moment problem aims to find a worst-case discrete distribution that satisfies a given set of moments. This paper studies the discrete moment problems with additional shape constraints that guarantee the worst-case distribution is either log-concave (LC) or has an increasing failure rate (IFR) or increasing generalized failure rate (IGFR). These classes are useful in practice, with applications in revenue management, reliability, and inventory control. The authors characterize the structure of optimal extreme point distributions and show, for example, that an optimal extreme point solution to a moment problem with m moments and LC shape constraints is piecewise geometric with at most m pieces. Using this optimality structure, they design an exact algorithm for computing optimal solutions in a low-dimensional space of parameters. The authors leverage this structure to study a robust newsvendor problem with shape constraints and compute optimal solutions.
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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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it