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Conformal Mixed-Integer Constraint Learning with Feasibility Guarantees
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Proposes C-MICL, which conformalizes learned constraint models inside mixed-integer constraint learning to give probabilistic ground-truth feasibility guarantees (at least 1-alpha) with only one or two trained models, evaluated on a membrane reactor regression task and a food basket classification task.
Problem
Solutions of mixed-integer constraint learning (MICL) optimization often violate the true underlying constraints because of model error and limited data, and existing fixes (ensemble-based W-MICL) lack formal guarantees and scale poorly with ensemble size.
Approach
Replace the learned constraint with a conformal prediction set (residual-normalized intervals for regression, logit-based conformal sets for classification) encoded as MIP constraints, so that feasible solutions are ground-truth feasible with probability at least 1-alpha under a conditional independence assumption, with formulation size independent of dataset size.
Key results
- C-MICL achieves the target >=90% ground-truth feasibility rate across all base models in the regression reactor case, while plain MICL and most W-MICL variants fall below it.
- In the regression case, C-MICL yields on average about 7% higher objective than heuristic baselines, and is roughly two orders of magnitude faster than W-MICL.
- In the food basket classification case, only C-MICL meets the 90% feasibility target, with objective differences around 1% and solve times comparable to a single MICL model.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use C-MICL instead of standard MICL | Membrane reactor optimization · Membrane reactor | ground-truth feasibility rate: C-MICL achieves ≥ 90%; none of the baseline MICL methods reach the desired coverage level | Helped |
| Use C-MICL instead of W-MICL | Membrane reactor optimization · Membrane reactor | ground-truth feasibility rate: C-MICL achieves ≥ 90%; W-MICL only empirically approaches the target when using neural networks | Helped |
| Use C-MICL instead of heuristic approaches | Membrane reactor optimization · Membrane reactor | objective value: 7% | Hurt |
| Use C-MICL instead of standard MICL or W-MICL | Food basket optimization · Food basket | ground-truth feasibility rate: C-MICL achieves ≥ 90%; only C-MICL satisfies the desired empirical coverage | Helped |
| Use C-MICL instead of W-MICL | Food basket optimization · Food basket | solution time | Helped |
| Use C-MICL instead of baselines | Food basket optimization · Food basket | relative optimality gap: around 1% | Mixed |
| Use C-MICL instead of MICL | Membrane reactor predictive model · Membrane reactor | MSE: 0.0597 -> 0.0841 | Hurt |
Surprising findings
- W-MICL only empirically approaches the target coverage when using neural networks, and the plain MICL baselines fail to reach the desired feasibility level regardless of model.
Limitations
- Conformal sets can become too wide and render the C-MICL problem infeasible when the underlying model performs poorly.
- Guarantee relies on a conditional independence assumption between feasibility and coverage that may fail if the feasible region is biased toward regions with poor coverage.
- Case studies restricted to MILP examples with ReLU NNs for classification, since OMLT only supports ReLU NNs for classification.
Experimental setup
Datasets: membrane reactor ODE simulations (1,000 samples, generated by authors) food basket dataset (5,000 baskets, Wiberg et al. OptiCL)
Benchmarks: membrane reactor design optimization food basket design optimization
Metrics: empirical ground-truth feasibility rate relative optimal objective difference CPU time MSE accuracy
Baselines: MICL W-MICL(5) W-MICL(10) W-MICL(25) W-MICL(50)
Builds on: Mixed-Integer Constraint Learning Wrapped MICL (W-MICL) Mondrian conformal prediction OMLT
Closest papers by meaning
- Learning Polyhedral Conformal Sets for Robust Optimization (NeurIPS 2026 · similarity 0.87)
- Conformal Prediction with Learned Features (ICML 2024 · similarity 0.85)
- Optimal transport-based conformal prediction (ICML 2025 · similarity 0.84)
- Conformal Validity Guarantees Exist for Any Data Distribution (and How to Find Them) (ICML 2024 · similarity 0.84)
- Smart Surrogate Losses for Contextual Stochastic Linear Optimization with Robust Constraints (NeurIPS 2025 · similarity 0.84)
- Co-optimization for Adaptive Conformal Prediction (NeurIPS 2026 · similarity 0.84)
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