Robust Optimization of Sums of Piecewise Linear Functions with Application to Inventory Problems
Bibliographic record
Abstract
Robust optimization is a methodology that has gained a lot of attention in the recent years. This is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models. Unfortunately, the second property is usually lost when the cost function that needs to be “robustified” is not concave (or linear) with respect to the perturbing parameters. In this paper we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set. Given that these problems are known to be intractable, we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed-integer linear program and relate this scheme to methods that are based on exploiting affine decision rules. Our new scheme gives rise to two tractable models that, respectively, take the shape of a linear program and a semidefinite program, with the latter having the potential to provide solutions of better quality than the former at the price of heavier computations. We present conditions under which our approximation models are exact. In particular, we are able to propose the first exact reformulations for a robust (and distributionally robust) multi-item newsvendor problem with budgeted uncertainty set and a reformulation for robust multiperiod inventory problems that is exact whether the uncertainty region reduces to a L 1 -norm ball or to a box. An extensive set of empirical results will illustrate the quality of the approximate solutions that are obtained using these two models on randomly generated instances of the latter problem.
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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".