Title: Conformal Mixed-Integer Constraint Learning with Feasibility Guarantees
Venue: NeurIPS 2025
Accepted: yes
Keywords: Constraint Learning

Conformal Prediction

Mixed-Integer Programming
Abstract: We propose Conformal Mixed-Integer Constraint Learning (C-MICL), a novel framework that provides probabilistic feasibility guarantees for data-driven constraints in optimization problems. While standard Mixed-Integer Constraint Learning methods often violate the true constraints due to model error or data limitations, our C-MICL approach leverages conformal prediction to ensure feasible solutions are ground-truth feasible with probability at least  $1{-}\alpha$, under a conditional independence assumption. The proposed framework supports both regression and classification tasks without requiring access to the true constraint function, while avoiding the scalability issues associated with ensemble-based heuristics. Experiments on real-world applications demonstrate that C-MICL consistently achieves target feasibility rates, maintains competitive objective performance, and significantly reduces computational cost compared to existing methods. Our work bridges mathematical optimization and machine learning, offering a principled approach to incorporate uncertainty-aware constraints into decision-making with rigorous statistical guarantees.


Peer reviews:
comments: SUMMARY:
In this paper, the author proposes a new method to address challenges in constraint learning(CL) in optimization problems. The authors introduces a framework that integrates conformal prediction to CL problems to ensure that solutions to optimization problems with learned constraints are practically implementable.

STRENGTHS AND WEAKNESSES:
Strengths:
-  The paper is well-written and well-organized.
-  The authors clearly define the problem, and the procedure to implement the proposed framework.
-  The authors validate their approach on both synthetic dataset and real dataset with real-world applications.


Weakness:
-  [Conditional independence assumption] The authors discuss the potential limitations of the assumption. While acknowledged as a necessary "theoretical compromise", the assumption is a notable one, and its potential impact warrants careful consideration by practitioners.

QUESTIONS:
[Influence of the specific conformal scores] In Section 4.1 and Section 4.2, specific conformal scores are chosen for the regression and classification problems, respectively. A more detailed analysis of how the choice of a conformal score function influences the results would be a valuable addition. Furthermore, how to evaluate or select between different valid score functions for a given application?

LIMITATIONS:
N/A
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Conformal Mixed-Integer Constraint Learning (C-MICL), a novel framework that integrates conformal prediction into mixed-integer constraint learning (MICL) to ensure probabilistic feasibility guarantees. The core idea is to replace heuristic approaches (e.g., ensemble-based W-MICL) with a principled conformal prediction-based mechanism that enables certified feasibility of solutions to learned-constraint optimization problems, under a mild conditional independence assumption. The method is model-agnostic and MIP-compatible, requiring only that the learned constraint model is representable as a mixed-integer program. It supports both regression and classification settings. Theoretical results are provided to justify the probabilistic guarantees. The empirical evaluation spans real-world-inspired MILP problems, including a chemical reactor optimization and a food basket design task. C-MICL consistently achieves the desired feasibility level, offers competitive objective performance, and exhibits significant speedups compared to ensemble methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The proposed C-MICL framework provides formal probabilistic guarantees on the feasibility of solutions with respect to the unknown true constraints, which is a major advance over existing heuristic approaches.\n\n2. C-MICL avoids ensemble-based methods and scales efficiently—requiring only one or two models—while achieving significant computational speedups without sacrificing objective performance.\n\n3. Across diverse case studies, C-MICL consistently achieves target feasibility levels, while maintaining competitive objective values.\n\n4. The paper is well-organized and clearly written, with intuitive explanations, theoretical results, and implementation details presented in a logical and accessible manner.\n\nWeaknesses:\n1. While plausible, the assumption that C-MICL feasibility and conformal coverage are conditionally independent (Assumption 4.1) is strong and difficult to validate in practice. This is the linchpin of the main guarantee, and its empirical verification is only approximate.\n\n2. In cases of poorly performing base predictors or high noise, conformal sets may become overly conservative, potentially leading to empty or trivial feasible sets. While discussed in the limitations, this could be a practical barrier.\n\nQUESTIONS:\n1.\tAs stated in the Limitation, in cases where the base model is inaccurate or uncertainty is large, the conformal set may become too wide, potentially making the C-MICL problem infeasible. Do the authors have strategies for detecting or mitigating this issue (e.g., fallback mechanisms, feasibility repair, or relaxed constraints)?\n2.\tWhile the paper claims generality beyond MILPs, experiments focus exclusively on linear settings. Have the authors tested C-MICL in mixed-integer nonlinear problems (MINLPs)? If so, what challenges arise?\n\nLIMITATIONS:\nThe authors have provided a thoughtful discussion of the limitations in Section 6, notably including the reliance on the conditional independence assumption, and the potential for conformal sets to become overly conservative when the predictive model is poorly calibrated. These are critical aspects and are appropriately acknowledged.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors discuss mathematical optimization problems for which a part of the constraint functions are not explicitly known. These unknown functions are replaced with predictive models using data. Although this idea has been studied in the literature before, the authors address the concern about the feasibility of the obtained optimal solutions, since these solutions may only satisfy the predictive model but not necessarily the true (unknown) mapping.\n\nUsing the core results from conformal prediction theory, the authors propose, what they call, Conformal Mixed-Integer Constraint Learning (C-MICL) method, which gives probabilistic guarantees on the learned constraint function and works with any model that can be represented as a mixed-integer program. To this end, the authors integrate conformal predictions in both a regression and a classification setting to ensure ground-truth feasibility with a certain probability. The authors compare their method on different metrics against multiple benchmarks on regression and classification datasets.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper is well-written. The overall message is clear, the mathematics is sound and well-defined.\n- Results look promising: Although other methods tend to find better optimal values, there is a clear improvement in ground-truth feasibility rate, and runtime when using the C-MICL method.\n\nWeaknesses:\n- Discussion on the conditional independence assumption is brief. The authors mention in the discussion that this assumption is reasonable. I would like to know earlier why this is a reasonable assumption and what situations this assumption does not hold. \n- All of a sudden, a secondary regression model comes into the picture in Section 4.1. It is not clear to understand its role, and why is it needed.\n- The authors report only the rate of the ground truth feasibility (I guess this is the number of feasible solutions among 100 instances). However, there is no evaluation of the magnitude of infeasibility. There could be many solutions that are obtained with other approaches which are very close to be feasible region, whereas C-MICL may result in a few but grossly infeasible solutions.\n\nQUESTIONS:\n- What happens when the feasible set for the outcome, Y is more complicated; e.g., it is defined by using integer variables and constraints?\n- Related to previous question, would the proposed approach still work if g function in (MICL) model also involves y?\n- In several places, the authors call the proposed approach model-agnostic. However, it does require predictive models to be MIP-representable, doesn't it?\n- In the experiments section, there are error bars, but it is not mentioned in the section what these bars represent exactly. Could the author elaborate on this when discussing the plots in the text? Moreover,  for the C-MICL method, the reported ground-truth feasibility rate does not always meet the target coverage level, as the error bars suggest. Could you elaborate why this happens, because the theory would suggest the target coverage level should be met? Is it because you are talking about empirical errors? Can the authors elaborate? \n- The authors claim Scalability and Efficiency concerning the size of the dataset. However, in your experiments, there seems to be a comparison with benchmarks for a fixed size of the dataset. For this dataset, the C-MICL method seems more efficient, and as it needs fewer models to train, the claim of scalability seems valid. However, there is no experimental support that this will hold for different dataset sizes, as with a fixed split for training and conformal calibration, the size of the dataset would affect all methods. Is this observation correct? Did the authors make comparisons for different dataset sizes? If so, could the authors elaborate on their claim regarding scalability?\n- What is the effect of the size of the data set used for conformal calibration? In one experiment, the authors use 20% of the data, and in the other, the authors use 8%. What is common practice here? How did you derive these different split ratios? And what effect would the size of the data set used for conformal calibration have on the discussed measures?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a methodology, called Conformal Mixed-Integer Constraint Learning (C-MICL), whereby conformal prediction is used to obtain probabilistic guarantees of feasibility on the solution to optimization problems with unknown constraints (where data can be used to learn the constraint). In particular, when the decision variables $x$ of an optimization problem must satisfy the constraint $h(x) \\in \\mathcal{Y}$ for unknown $h$, the authors propose to construct a conformal prediction set $\\mathcal{C}(x)$ which, under standard exchangeability assumptions, contains the value $h(x)$ with probability $1-\\alpha$ for some chosen $\\alpha$. Then, the set containment constraint $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ can be used as a (conservative, but valid with high probability) replacement for the original unknown constraint, and the the feasible set of the new problem will be feasible for the original constraint with probability $1-\\alpha$. The authors describe how this framework can be used in both the regression and classification settings, and run extensive experiments demonstrating that their approach yields better feasibility rate when compared to previous methods, while remaining computationally tractable.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- This is a very interesting problem setting, and the application of conformal prediction to this problem is a valuable insight. \n- The paper is clearly written and easy to follow. \n- The experiments are extensive and well-documented. \n\nWeaknesses:\n- There are a number of papers in the recent literature that have looked at integrating conformal prediction into various forms of optimization problems to deal with robust constraints and/or objectives. In the related work, the authors cite references [46] and [47], but it might be worth discussing the current work in the context of the broader set of papers at this intersection, including, e.g.:\n  1. Johnstone and Cox, [Conformal Uncertainty Sets for Robust Optimization](https://arxiv.org/abs/2105.14957)\n  2. Kiyani et al., [Decision Theoretic Foundations for Conformal Prediction: Optimal Uncertainty Quantification for Risk-Averse Agents](https://arxiv.org/abs/2502.02561)\n  3. Patel et al., [Conformal Contextual Robust Optimization](https://arxiv.org/abs/2310.10003)\n  4. Yeh et al., [End-to-End Conformal Calibration for Optimization Under Uncertainty](https://arxiv.org/abs/2409.20534)\n\n- Assumption 4.1, on the conditional independence of the feasibility of C-MICL and conformal coverage, seems strong, and not obviously true. \n- In cases where the function $h$ itself is not random, replacing the constraint $h(x) \\in \\mathcal{Y}$ with the conformal set containment $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ seems like it would be conservative in general.\n\nQUESTIONS:\n- The conformal set containment constraint $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ seems like a challenging constraint to enforce in general, depending on the structure of $\\mathcal{C}(x)$. In Sections 4.1 and 4.2 of the paper, you show how various regression and classification settings can yield conformal sets whose containment problems are MIP-representable, but are there other settings where such a formulation will not be easily obtained, and where this constraint will be hard to enforce in a MIP?\n- If I understand correctly, assumption 4.1 is saying you assume that, if $h(x) \\in \\mathcal{Y}$, then the event $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ is independent of the event $h(x) \\in \\mathcal{C}(x)$. Whether or not this assumption holds seems critical for the validity of the probabilistic feasibility guarantees. Is this a reasonable assumption in practice? If so, can you give formal, mathematical intuition for why this is an assumption that is likely to hold? I am willing to raise my score if provided a convincing answer to this question.\n- The statement in lines 199-200 that \"we assume that whether a point satisfies the constraint $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ does not affect the probability that the conformal set contains the true function value\" seems imprecise, given that the exact place where this is invoked in the proof includes the entire feasible set $\\mathcal{F}_N$ of the C-MICL problem, not just the set containment $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$.\n- It seems that replacing the constraint $h(x) \\in \\mathcal{Y}$ with the conformal set containment $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ is conservative, as it results in larger objective values (seen in Figures 2 and 5). How do these values compare with the ground truth problem's optimal objective value?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}