A Tractable Approach To Inverse Optimization Under Euclidean Norm
Bibliographic record
Abstract
The conventional optimization assumes that the problem and its parameters are known, and it utilizes this information to determine the optimal solution. Inverse optimization works in reverse by determining different parameters of an optimization model such that a given dataset of observed decisions from the past becomes optimal for the model. The parameters imputed through inverse optimization can be in the objective function and/or the constraints of the model. When inferring the constraint parameters, the choice of objective for the inverse optimization problem can result in different inverse optimal solutions. However, it is unclear which solution provides the best fit to the data. In this study, a goodness-of-fit measure is first introduced to evaluate the fit between the model and data and determine the quality of the inferred feasible region based on the distances of data points from its boundary. Next, employing this measure as the objective function, a multi-point inverse optimization problem under the Euclidean norm is proposed to infer the feasible region of a linear optimization model. Given the nonlinear nature of the Euclidean norm, a relaxation technique using the non-smooth L1 penalty function is proposed for the inverse optimization problem. This reformulates the non-convex mixed-integer quadratically-constrained programming problem into a mixed-integer quadratic programming problem which is more tractable. Then, an exact heuristic method and a greedy heuristic method are introduced to alleviate the computational challenges of the problem. Finally, two application examples to illustrate the practicality and effectiveness of our proposed model and solution approach are presented. In the first application, our model determines the implicit criteria based on which a patient is identified as an outpatient without requiring hospital supervision. The second application focuses on improving the recommended diets by uncovering hidden preferences and suggesting meal plans based on individuals' past food choices.
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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 | 0.005 | 0.012 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".