Machine Learning–Augmented Optimization of Large Bilevel and Two-Stage Stochastic Programs: Application to Cycling Network Design
Bibliographic record
Abstract
Problem definition: A wide range of decision problems can be formulated as bilevel programs with independent followers, which, as a special case, include two-stage stochastic programs. These problems are notoriously difficult to solve, especially when a large number of followers are present. Motivated by a real-world cycling infrastructure planning application, we present a general approach to solving such problems. Methodology/results: We propose an optimization model that explicitly considers a sampled subset of followers and exploits a machine learning model to estimate the objective values of unsampled followers. We prove bounds on the optimality gap of the generated leader decision as measured by the original objective function that considers the full follower set. We then develop follower sampling algorithms to tighten the bounds and a representation learning approach to learn follower features, which are used as inputs to the embedded machine learning model. Through numerical studies, we show that our approach generates leader decisions of higher quality compared with baselines. Finally, in collaboration with the City of Toronto, we perform a real-world case study in Toronto, where we solve a cycling network design problem with over one million followers. Compared with the current practice, our approach improves Toronto’s cycling accessibility by 19.2%, equivalent to $18 million in potential cost savings. Managerial implications: Our approach is being used to inform the cycling infrastructure planning in Toronto and can be generalized to any decision problems that are formulated as bilevel programs with independent followers. Funding: This work was supported by City of Toronto Transportation Services and the Natural Sciences and Engineering Research Council of Canada [NSERC Alliance Grant ALLRP 561212-20]. Supplemental Material: The electronic companion is available at https://doi.org/10.1287/msom.2024.1317 .
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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 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 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".