Subproblem Approximation in Dantzig-Wolfe Decomposition of Variational Inequality Models with an Application to a Multicommodity Economic Equilibrium Model
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Bibliographic record
Abstract
We present a modification to Dantzig-Wolfe decomposition of variational inequality (VI) problems that allows for approximation of the VI mapping in the subproblem. The approximation is parameterized by the most recent master problem solution, and it must satisfy two simple requirements. In an electronic companion (online appendix), we show that the proofs of convergence and other important properties go through with subproblem approximation. The approximation procedure is illustrated by an application to a class of multicommodity economic equilibrium models (MCEEMs): the standard Dantzig-Wolfe decomposition by commodity does not allow the subproblem to be decomposed into separate subproblems for each commodity, but we show two ways to approximate the subproblem's inverse demand function, and both ways allow the subproblem to be broken into separate single-commodity problems. A further approximation is combined with each of the inverse demand approximations; in effect, an approximate supply or demand curve is introduced into each commodity's subproblem for transfers of commodities between different subproblems, thus allowing the subproblems to produce better proposals. An illustration is included for an MCEEM that represents energy markets in Canada.
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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 | 0.000 |
| 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 | 0.002 |
| Open science | 0.001 | 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