Conformal Mixed-Integer Constraint Learning with Feasibility Guarantees
Authors: Daniel Ovalle
Lorenz T. Biegler
Ignacio E Grossmann
Carl D Laird
Mateo Dulce Rubio
Venue: NeurIPS 2025 spotlight ()
Keywords: Constraint Learning
Conformal Prediction
Mixed-Integer Programming
Primary area: optimization
TL;DR: We propose Conformal Mixed-Integer Constraint Learning, a novel framework that provides probabilistic feasibility guarantees for data-driven constraints in optimization problems.

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.

Conformal Mixed- Constraint Learning with Feasibility Guarantees
Daniel Ovalle
Lorenz Biegler
Department of Chemical Engineering Carnegie Mellon University
Department of Chemical Engineering Carnegie Mellon University 1b01 @011aan cmu.edu
dovallev@andrew cmu. edu
Ignacio E. Grossmann Department Chemical Engineering Carnegie Mellon University ig0c0andrew. cmu. edu
Carl D. Laird
Department Chemical Engineering
Carnegie Mellon University
claird@andrev. cmu edu
Mateo Dulce Rubio Center for Data Science New York University mateo. d@nyu. edu
Abstract
We propose Conformal Mixed Mixed-Integer Constraint Learning (C-MICL), a novel framework that provides probabilistic feasibility guarantees for data-driven straints in optimization problems. While standard Mixed-Integer Constraint Learn- ing methods often violate the true constraints due to model error data limitations, our C-MICL approach leverages conformal prediction to ensure feasible solutions are ground-truth feasible with probability least 1-a, under a conditional in- dependence assumption. The proposed framework supports both regression and classification tasks without requiring access to the true constraint function. while heuristics. Experi-
avoiding the scalability issues associated with ensemble-base ments on real-world applications demonstrate that C-MICL consistently achieves target feasibility rates, maintains competitive objective performance, and signifi- cantly reduces computational cost compared existing methods. Our work bridges mathematical optimization and machine learning, offering principled approach incorporate uncertainty-awar constraints into decision-man making with rigorous statistical guarantees.
Introduction
Constraint Learning (CL) deals with inferring the functional form of constraints or objectives from observed data embed them into mathematical optimization problems, particularly when parts of the system behavior are difficult to model explicitly [1]. This is especially relevant in settings where the relationship between decision and output variables is unknown but can be learned from data, enabling a supervised learning approach [2, 3]. It is also valuable in surrogate modeling scenarios where the relationship is known but difficult optimize, either due its nonlinearity, computational cost, or complexity [4-6]. That is, CL. simply leverages modern machine learning systems to learn the structure of complex constraints directly from data, enabling prediction of constraint boundaries at unobserved points to efficiently navigate feasible spaces and identify optimal solutions.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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The core principle that enables CL is that the predictions of many machine learning models can be formulated as closed-form algebraic expressions that are admit exact mixed-integer programming (MIP) representations [1, 7-17]. This integration is com- monly referred to as Mixed-Integer Constraint Learning (MICL.. However, a critical challenge arises from the very nature of optimization algorithms: they systematically leverage the mathematical structure of the problem to identify extreme points. As a result, small approximation errors in the learned constraint model (e.g., arising from noisy data, model bias, insufficient training) can be systematically exploited during the optimization. This behavior often leads to solutions that are infeasible with respect to the true system, meaning that the optimal decision variables found by optimization programs under CL frequently violate the original constraints when evaluated using the true underlying model [18]. This lack of true feasibility results in solutions that cannot be implemented in practice, or that are overly conservative due to the need for additional safety margins.
and commonly
To improve the robustness of MICL solutions, recent research has proposed ensemble-based ap- proaches where multiple predictive models are trained on different bootstrap samples of the data [19]. In particular, Maragno et al. [7] introduce framework in which /' models are independently trained on bootstrapped datasets, and the optimization constrained so that a 1-a fraction of these models must satisfy the imposed constraints. There is some empirical evidence that suggests that this heuristic yields solutions that are typically feasible with respect to the true underlying system, and its robustness improves as P increases. However, this approach lacks formal guarantees as there is no assurance that the final solution will satisfy the true constraints with high probability, particularly in the presence of model misspecification or low-quality data used in the MICL step. Moreover, the method suffers from key scalability issues: as P grows, the size of the optimization problem increases accordingly due the replication of model constraints and auxiliary variables, rapidly making the problem computationally intractable for large ensembles.
In this paper, we introduce Conformal Mixed-Intege Constraint Learning (C-MICL), novel model- agnostic framework for incorporating data-driven constraints into optimization problems with formal probabilistic feasibility guarantees. Our method supports both regression and classification settings and is computationally efficient without sacrificing significant solution quality. Specifically:
Probabilistic Guarantees: We provide new optimization formulation that guarantees ground-truth feasibility with probability at least 1-a when the true constraint function is unknown or inaccessible, under mild conditional independence assumption.
Model-Agnostic and MIP-Compatible: C-MICL is compatible with any predictive model that admits a MIP encoding, enabling integration into standard optimization solvers.
Scalability and Efficiency: Our method avoids query access or model refinement during optimization, does not scale with dataset size, and requires training at most two models, achieving ordgrs-of-magnitude computational speedup over ensemble-based heuristics.
Empirical Validation:
We demonstrate the effectiveness of C-MICL on real-world case studies in both regression and classification settings. Extensive experiments show that our method consistently achieves target feasibility levels, while delivering competitive objective values with significantly reduced computation times compared to state-of-the-art baselines.
2 Literature Review
Mixed-Integer Constraint Learning. The mixed-integer constraint learning framework has en- abled a wide range of developments in machine learning, including neural network verification [20-27], adversarial example generation [28-30], and applications in reinforcement learning [31-33]. It has also been successfully applied to domain-specific tasks in power systems [34, 35], healthcare [36, 37], supply chain management [38, 39], and chemical process design [40]. For a comprehensive overview of recent advances in constraint learning methodologies, we refer readers to Fajemisin et al. [1], and for review focused on engineering applications, see Misener and Biegler [41]. How- ever, existing CL frameworks generally lack guarantees of true system feasibility, particularly in settings where predictive uncertainty from noisy data or model misspecification is prevalent but typically ignored. The approach proposed in this work directly addresses this gap by providing formal probabilistic feasibility guarantees for uncertain learned constraints embedded into optimization problems.

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Trust-Reginn Muust-Region-Filter for Optimization. Trust region filter methods have been proposed as a strategy to optimize over learned constraints while ensuring that the solution aligns with that of the true, underlying system [18]. These approaches operate by iteratively refining the learned constraint through local sampling and evaluation of the truth function, progressively narrowing the optimization domain within trust regions where the learned model is accurate [42, 43]. However, such methods rely on strong assumptions: they typically assume noise-free data, differentiability of the learned constraint, and direct query access to the truth function. In contrast, our framework is designed for settings where these assumptions do not hold. We account for noisy data, allow for non-differentiable learned models, and, most importantly, do not require access to knowledge of the underlying function, enabling optimization with high-probability feasibility guarantees even under limited and imperfect information.
manformal-Enhancee Optimization. Building on advances in conformal prediction [44-46], recent work has explored its integration with optimization. Lin al. [47] proposed conformalized inverse optimization, targeting the estimation of unknown parameters embedded in the objective or constraints of optimization problem, and providing uncertainty quantification parameters. Our approach addresses a different problem, namely learning and conformalizing the functional forms of constraints themselves, without relying on fixed parametric structure, which allows for more expressive modeling of complex or unknown system behaviors. Separately. Zhao al. [48] introduced conformalized chance constrained optimization, in which constraints involving random variables are required to hold with high probability. Their method encodes quantile thresholds from conformal prediction directly into the optimization formulation, which scales with the number of data points. Unlike this approach, ours is focused on uncertainty of model predictions rather than model stochasticity, and supports learning general constraint functions.
these learned
Other related efforts include Johnstone and Cox [49], who construct conformal ellipsoids for robust optimization, and Yeh al. [50], who learn convex uncertainty sets for robust objectives using differentiable conformal layers. In contrast, C-MICL conformalizes feasibility regions rather than robustifying objective values. Similarly, Kiyani al. [51] and Patel et al. [52] employ conformal sets for utility-aware or risk-sensitive decision-maker output quality, whereas C-MICL focuses on constraint satisfaction under uncertainty from model estimation. Our framework supports general, nonconvex feasible regions and provides distribution- free probabilistic feasibility guarantees from fixed-size optimization formulations regardless of dataset size, allowing practitioners to leverage large datasets for improved model accuracy without inflating computational cost. A key distinction of C-MICL is that it centers on feasibility-aware decision-making rather than on robustifying objectives or actions for downstream optimization problems.
optimizing value-at-risk or ensuring robust
Preliminaries
3.1 Problem Set-Up
Consider a constrained optimization problem over set of decision variables (x. 2) where certain constraints depend unknown or difficult-te mo-dell functions relating decision variables to outcomes We assume no access to the true underlying mapping denoted by l(x) = EΙy X = x]. Instead, we have access to a dataset Disin = {(z,, yı) }. where each observation (x,, y:) is drawn independently from common (unknown) underlying distribution Pxy. Note that z are additional decision variables that may constrain x, but do not influence through h((z).
The goal of Constraint Learning is to approximate h(x) with a predictive model h(x) trained on the observed data Dmmin- Once the model h(x) is trained, we embed it in mixed-integer optimization program, defining the following Mixed- Integer Constraint Learning (MICL) problem [1]:
min z.y.=
(x,
s.t.
g $
h(x) = y
(MICL)
22, E X
y ye y
3

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where fzz 2) is known real-valued objective function and g(x, 2) encodes set of known constraints. Moreover, X S R x n defines the feasibility set for the decision variables (x, 2) where both and can take both real and integer values. Finally, y defines the feasible set for the outcome variable y, where feasibility must be satisfied with respect to the true (but unknown) function h(x)).
The formulation above captures a general form of a mixed-integer constraint learning problem. Specific problem instances correspond to different definitions of the feasible prediction set y. In the regression setting, J is any subset of R that is representable within a MIP framework. For instance, in our experiments, we define y = lg. y). which enforces lower and upper bounds on the continuous output variable y. In the multi-label classification case, y = K is a subset of the set of possible labels K. = Kdde U ICund, which we assume partitions into disjoint sets of desired classes KÇdes and undesired classes XCund
By construction, all feasible solutions (z',=") from the optimization problem MICL are feasible with respect to the learned model h(x),, meaning that h (a²) y. However, since h(x) is unknown and only approximated via f(z), ground- truthth-rrhhhha feasibility with respect the true function h(x") is not guaranteed. We formalize this conceptual difference in the following definition.
Definition 3.1. (Ground- Truth Feasibility) Given a feasible solution (a', 2²) to the MICL optinizaai problem, the solution is said achieve ground-trund feasibility if y (x') E Y, where h(z) the true constraint function.
Recent research by Maragno, Wiberg, Bertsimas, Birbil, den Hertog, and Fajemisin [7] proposes heuristic approach called Wrapped model MICL (W-MICL) to address the challenge of ensur- ing ground-truth feasibility of MICL, solutions. W-MICL trains an ensemble of P models using independent bootstrapped datasets rather than relying on single predictive model. Each model (x)) provides a distinct prediction, collectively capturing model uncertainty. Within the MICL optimization formulation, the constraint M(x) E y is replicated for each trained model, and feasibility (x) E Y is enforced for (1-a) fraction of them, implemented via standard big-M formulations in MIP [7]. While empirical results suggest that W-MICL improves the ground-truth feasibility of optimal solutions, the method remains a heuristic and does not offer formal feasibility guarantees.
Remark 3.1. We assume that the underlying predictive model π(x)) admits a reformulation as mixed-intege program (MIP), such as Neural Networks with ReLU activations (ReLU NNs) [8- 10],
Trees (GBTs) [11, 12], Random Forests (RFs) [13-16], or Linear-Model Decision Trees (LMDTs) [17]. Then, the MICL problem is itself MIP that can be solved using standard branch-and- bound or global optimization algorithms J53, 54J. While details on embedding these models into MIPs are provided in Appendix A, we emphasize that our formulation remains model-agnostic, requiring only that the predictive model h(x) is MIP -ie-rrerreeettbbe..
3.2 Conformal Prediction
Conformal prediction is a statistical framework for model-free uncertainty quantification that provides probabilistic guarantees for prediction sets constructed from any given (fixed) predictive model. This framework has gained significant popularity as an assumption-lean model-agnostie computationally efficient method offering rigorous statistical coverage guarantees without requiring model retraining or modifications pre-trained machine learning systems [55-57].
Formally, suppose we have access to a calibration dataset of N data points (11.))),.... (NN, UN) drawn exchangeably from the underlying distribution Pxy. Given a new data point (IN+1, +1) drawn exchangeably from Pxy. conformal prediction provides a principled method for constructing a conformal set C. (æN+11 containing the true label yN+1 with probability at least 1 a, without making additional assumptions about the underlying distribution beyond exchangeability.
The core idea behind conformal prediction is to compute conformal scores {s(2;, y1) ) that capture model uncertainty over the calibration data. For instance, in regression settings, common conformal score function corresponds to model absolute residuals s(x, y)
1y h(z)). The
If y a vector-valued outcome, [y, T] imposes elementwise constraints on y V.
A random vector (Zi...., ZN) is exchangeable if (Zo()).... N)(N)) follows the same distribution for all permutations of the indices (L...., NN [57]. For instance, i.i.d. random variables are exchangeable.

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conformal prediction set is then defined as:
C. (TrN+1) = {v ((xN+119) S <Q
where T1-a is the (1-a)(1 + 1/N) -quantile of the empirical distribution of conformal scores {s(21, v1),.................. s(EN, yN)). This construction yields the following key marginal guarantee.
Theorem 3.1 (Theorem 3.2 in [57]). Assume that (X1, Yi)..... ..... (XN+1, YN+1) exchangeable random variables. Then, for any aser-specified coverage level a E (0, 1). we have that the conformal set set from eq. (1) satisfies
P P(N++1 E (XN+1)))) 2 a.
Recent advancements have extended the framework to construct adaptive prediction sets with conditional coverage guarantees, whose size varies according to the model's confidence specific inputs, producing narrower intervals in regions of low uncertainty [58, 59]. This adaptive property particularly relevant our work, as informs optimization algorithms about local prediction confidence when checking for feasibility. For instance, Mondrian conformal prediction [57, 60] enables conditional coverage guarantees for both ground-truth feasible points {a: -h(r) E ) and infeasible points {x h(x) & )). This conditional coverage guarantee is formalized in the following Lemma, with the proof and details on Mondrian conformal sets deferred to Appendix B.I.
Lemma 3.1. (Ground- Truth Feasibility Conformal Coverage Guarantee): Assume that (X1,Yi).... Yi).... ,.....,(NN-L (XN-1. Y are exchangeable random variables and that s s(X, ) is a symmetric score finnction. Mondrian conformal sets satisfy the conditional coverage guarantee at level ac:
P(YN+1 CCa(NN+)) YN+1 E y) a. PPYN+1 E (XN+))) YN+1 $ y) 2 a.
Intuitively, this condition ensures that coverage holds simultaneously within both the feasible and in- feasible regions, by constructing conformal sets calibrated separately for each group. This conditional guarantee is strictly stronger than (and implies) the marginal property from Theorem 3.1, preventing the procedure from achieving marginal coverage when high coverage on one subset compensates for poor coverage on the other.
Conformal Mixed-Integer Constraint Learning
In this section, we introduce proposed Conformal Mixed-Integer Constraint Learning (C-MICL) framework, integrating conformal prediction into MICL problems to provide probabilistic guarantees on the ground-truth feasibility of feasible solutions. The core idea of C-MICL is to conformalize the
learned constraint h(x) e thereby constructing statistically valid uncertainty set that the predictions must satisfy. The general C-MICL problem can be written as follows:
min 2,2 s.t.
f(r, 2)
g(I, 2) < 0
(C-MICL)
(x, E X'
C. (x 5
where (x) is conformal set for the decision variable based on the learned model f((z), target coverage level a and a calibration set Deat- In the following, we detail how construct C, (x) for both regression and classification settings, and how the constraint C.. (x) C Y can be formulated using MIP. Importantly, our approach remains compatible with the existing definition of ground- -ruth feasibility provided in Definition 3.1, h(xx) y, requiring modifications to the underlying feasibility notion.
Remark 4.1. The feasible regions defined by the oracle constraint h(x) y and the conformal set containment C. C do not exhibit clear inclusion relationship in general. In particular, neither region necessarily contains the other, and therefore one is not more conservative nor yields larger/smaller optimal values in general.
We assume that the model h(x) is trained on Diran and considered fixed thereafter. The conformal sets are then constructed using a disjoint calibration dataset Deat of size N. This approach preserves the theoretical coverage guarantees from Theorem 3.1 and Lemma 3.1 without requiring model retraining [57]. To establish formal guarantees for the C-MICL approach, we introduce the following assumption about the conformal coverage of the feasible region of the C-MICL problem.
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Assumption 4.1. (Conditional Independence of Feasibility and Coverage): The events of C-MICL feastbility and conformal coverage are conditionally independent given ground-trutt truth feastbility.
This assumption states that the C-MICL constraints do not systematically bias the coverage properties of conformal prediction sets, conditional on the true (unknown) feasibility status of ((x) That is, we assume that whether point is C-MICL feasible, i.e., (x', z') FN {(z, 2) X' g(z, 2) S 0, (x) C }} does not affect the probability that its conformal set contains the true function value, h(x²) E (z'), given its ground-truth feasibility (h(x') E Y) or infeasibility (h(x') $ y). We provide a detailed discussion and interpretation this assumption in Appendix B.3.
Under this condition and ground-truth feasibility conformal coverage guarantee (Lemma 3.1), we establish our main theoretical result the following theorem, which we prove in Appendix B.2. This result provides probabilistic certificate for the ground-truth feasibility of feasible solutions to the C-MICL problem.
Theorem 4.1. Let FN = {(x,2) X: g(x,2) S (x) y} be the feasible region of the C-MICL problem. Under the same conditions of Lemma 3.1 and Assumption 4.1, if FN is nonempty, then any feasible solution (x',z)) FN is ground-truth feasible with probability least 1-a:
P (hh(z') EY (d' E FN) 2
4.1 Regression Conformal MICL
For the regression setting, we adopt two-step procedure to construct predictive model h(x) and then quantify its uncertainty through an auxiliary uncertainty function û(r). Specifically, in the first step we use Duain to train predictive regression model h(z). Then, we use the set of absolute residuals of the model in the training set Dies = {(z,, (h(x,) well ) to train the secondary regression model ú(z), which estimates the prediction uncertainty of f(xz) as a function of the input variable x [61]. While alternative one-step approaches exist for simultaneously estimating both prediction and uncertainty models (often by leveraging model-specific choices of hxx)), we adopt this two-step procedure because it is simple, broadly applicable, and model-agnostic. For a discussion of joint training techniques, we refer the reader to Angelopoulos et al. [57]. Note that our framework imposes no restrictions on the choice of predictive model for ((r); however, to ensure integration with the optimization formulation, assume that ú(z) admits a MIP representation.
Although ((x) provides a heuristic estimate of uncertainty, it can still suffer from misspecification due to data noise, bias, or limited training. To address this and provide formal guarantees on predictive reliability, we apply conformal prediction to calibrate the uncertainty estimates. Specifically, for
each point in the calibration dataset Deat, conformal score is computed s(x;, vi) |h(x.)-wl following the procedure in Lei al. [62]. Then, as outlined in Section 3.2, the conformal quantile Q1-a Quantile (s( (11, 31),...,,((, S(NN ) (1 a1 (1 + 1/N)) is computed based on the set of conformal scores. Using this quantile the conformal set constraint can be expressed as:
M(xr,)
(x) cy + (Mxx + ύi ((x) C [y
This set is defined by pair of algebraic expressions that further constrain the lower and upper bounds on y and can be directly embedded into the optimization formulation (see Appendix C). In summary, C-MICL in the regression setting involves (i) training an uncertainty model ((x), (ii) computing the conformal quantile offline (so that the formulation size remains independent of N), and (iii) expressing the valid region for y through MIP constraints in terms of h(x)) and ū(x))
4.2 Classification Conformal MICL
For the classification setting, C-MICL does not require integrating an additional uncertainty model into the formulation. Instead, score functions can be directly computed from the predictor h(x;) = P(Y X = 2)., which typically involves a nonlinear transformation such as softmax [55]. However, in MICL, it common practice to remove the final nonlinear transformation and operate directly on the pre-activation values of the last layer, commonly referred to as logits, to preserve linearity [8]. This approach does not affect the classification result: since softmax and sigmoid functions are monotonie, the class with the highest logit still corresponds to the highest probability.

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To enable integration into MIP, we propose a valid conformal score function that operates directly in logit space. Specifically, we define the conformal score as:
(z), v.)
Σ y h (2x) KEK
(5)
where we select the negative of the logit output corresponding to the correct class, assuming that yi is one-hot encoded. This formulation preserves the validity of the conformal method while enabling exact and efficient representation within linear optimization model.
Given 91-a Quantile (s( v1)...., s(ZN, i UN); (1 a) 1/N]) defined in Section 3.2, we define the conformal sets in the classification setting as:
(x) CJ - -h(x)* Qi-a
Vk K.nnd
(6)
which ensures that only desired classes k E Kides are included in the conformal prediction set, by retaining those whose conformal scores lie below the quantile threshold. This behavior can be captured within a MIP framework by introducing an indicator constraint of the form 1 [ -fh(1)*** < (h-a) for all E K. and enforcing it to be active only for desired classes k E Kales, This logical constraint can be further reformulated as a set of algebraic mixed-integer inequalities using standard big-M techniques; we refer the reader to Appendix C for further details on this reformulation.
Remark 4.2. The functional forms chosen for the score functions s(x) and feasibility sets y serve primarily to illustrate the algorithmic implementation of C-MICL across broad classes of regres- sion and classification problems within optimization formulations. However, any alternative score functions (see (57]) and feasibility sets can be used, provided they admit MIP representations [41].
5 Computational Experiments
We empirically validate our Conformal MICL (C-MICL) approach in both regression and multi-class classification settings. In each case, we benchmark against standard single-model MICL methods as well as the ensemble-based heuristic W-MICL proposed by Maragno al. [7]. We denote the latter as W-MICL(P), where P' refers to the ensemble size. For both settings, we evaluate performance on 100 randomly generated optimization instances, each defined by sampled cost vector. This approach considers the calibration set Deat as fixed and allows to sample from the feasible region for each C-MICL formulation. This allows us to empirically validate the theoretical guarantee in Theorem 4. suggesting ground-truth feasibility of optimal solutions that are fcasible by construction. Moreover, in the Appendix we empirically verify that the assumptions from Theorem 4.1 approximately holds in our experimental settings for both regression and classification problems.
In the following, report: (i) the empirical ground-truth feasibility rate, (ii) the relative distance of optimal value A%, = fi- -CxIL 100% for each baseline method i when compared to ours, and (iii)
Séamu
the average CPU time required to solve each instance. The results shown in the main text correspond to a target coverage level of a = 10% across 100 runs with 95% confidence intervals (CIs) detailed in Appendix E. Additional results for a = 5% are available in Appendix E. and E.2. Key architectural and hyperparameter choices are summarized in the main text, while full details on hyperparameter selection, training, and optimization formulations are provided in Appendix D.
We focus our case studies on iixedeinteger linear programming (MILP) examples, which are faster to solve in practice. This choice is made without loss of generality. as all proposed formulations and theoretical guarantees remain valid for both mixed-integer linear and nonlinear programs. All optimization problems were solved using the MILP solver Gurobi v12.0.5 with relative opti- mality gap of 1% [63]. Machine learning models were implemented using scikit-learn and PyTorch, and subsequently integrated into Pyomo-based [64] optinization formulations via the open-source library OMLT [65]. The code required to reproduce all numerical experiments presented this section is publicly available in our GitHub repository. which additionally includes detailed tutorials on conformal prediction and its integration into Pyomo-based optimization formulations.
'Specifically, we implement the formulation proposed Section 3.1 Maragno [7] using a bootstrap sample proportion 0.5 as done by the authors. 'GitHub: https: // //gi thub. com domm/ddoalle//cnnll micl
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5.1 Regression setting
We consider the optimal design and operation of membrane reactor for methane aromatization and hydrogen production following Carrasco and Lima [66]. The goal is to determine five input variables (reactant flows, operating temperature, and reactor dimensions) ensuring a product flow 2 50, defining the feasible region y. The true system behavior is modeled by set of ordinary differential equations serving the oracle h(X). In this setting, y [0, 100] defines an underlying regression problem. To generate training data, we sample 1,000 input points from X' and evaluate y via the discretized ODEs with added Gaussian noise to simulate measurement uncertainty [67]. See Appendix D for additional details.
minimize operational cost while
We train baseline MICL models using the full dataset, following established methodologies for different predictor types: ReLU NNs [9], GBTs [11], RFs [16], and LMDTs [17]. To evaluate the performance of the robust ensemble heuristic W-MICL, we follow the approach of Maragno al. [7], training ensembles size P E {5. 10, 25, 50} using the same cross- validate hyperparameters as their corresponding single models described in Appendix D.1. Finally, we implement our proposed C-MICL method, which only requires training two models: h(x) and ū(x:). We use the same model architectures and hyperparameters as the baselines, allocating 80% of the data for training, and the remaining 20% for conformal calibration. All base models shared common architecture for the uncertainty model: ReLU NN with two hidden layers, each with 32 units.
Figure 1 presents the main results, comparing the ground truth feasibility coverage of the various methods across these instances for a = 10%. The figure shows that, across all underlying models, our proposed C-MICL method consistently achieves the target ground-truth feasibility rate, guaranteed by Theorem 4. In sharp contrast, none of the baseline MICL methods reach the desired coverage level, regardless of the machine learning model employed. Moreover, the W-MICL heuristic from Maragno et al. [7] only empirically approaches the target coverage when using neural networks. These results highlight the importance of theoretical guarantees: in settings where reliable machine learning model is available to support MICL or W-MICL approaches, our Conformal MICL framework ensures ground-truth feasibility by construction. Furthermore, the level of probabilistic guarantee can be flexibly controlled by the parameter a, allowing practitioners to tune the method to any desired confidence level in (0, 1) (with violin plots of the true constraint values h(x) for both a levels illustrate the magnitude of violations, showing that C-MICL not only incurs the fewest violations but also achieves the smallest violation magnitudes when they occur.
0.05 included in Appendix E). the same appendix,
Figure 2 presents the relative distance in optimal objective value between each method and proposed C-MICL approach, while Figure reports the average CPU time (in seconds) required solve the 100 optimization instances. In both figures, lighter-colored bars denote methods that failed to meet the desired feasibility rate shown in Figure 1. Results correspond to ReLU NNs, which exhibited the strongest empirical performance terms of feasibility.
Ground-Truth Feasibility Rate
1.0
1
0.8
0.6
0.4
ReLU NN
Boostng
0.2
RandomForest
LinearModeDDT
0.0
MICL 16,11
W-MICL/5) [7]
W-MICL(10)
W-MICL(25) [7] W-MICL(50) [7]
C-MICL (Ours)
Figure 1: Empirical ground-truth feasibility rate of optimal solution across MICL methods on
100 optimization problem instances. Our Conformal MICL approach (rightmost bars) consistently achieves the target ground-truth feasibility rate 2 90% regardless of the underlying hase model used. Previous methods demonstrate variable performance below this theoretical guarantee. Error bars correspond to 95% confidence intervals for ground-truth feasibility rates across 100 runs.

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C-MICL (Ours)
W-MICL(50) [7]-
104
W-MICL(5O) [7]
W-MICL(25)
W-MICL(25) [7]
W-MICL(10) [7]
W-MICL(10) [7]
W-MICL(S) [7]
101
W-MICL(S) [7]
MICL [9]
MICL [9]
-20
-10
10 20
10 2
10
1
10
10a
4% Optimal Value w.r.t. C-MICL (Ours)
Time [s] to Obtain an Optimal Solution
Figure 2: Percentage difference in optimal value from NN-based MICL methods compared to C- MICL. Results show an average 7% reduction in objective value compared to heuristics, indicat- ing a modest trade-off for feasibility guarantees. MICL (lighter interval) does not achieve the de- sired ground-truth feasibility rate. We present 95% confidence intervals across 100 runs.
Figure 3: Average computational time to find optimal solution from NN-based MICL methods. Our C-MICL approach (top) is two orders of magnitude faster than previous methods. MICL (lighter bar) does not achieve the desired empir- ical feasibility rate. Error bars represent 95% confidence intervals across 100 runs.
Figure shows that our proposed C-MICL method achieves the desired feasibility guarantees with only modest average reduction of 7% in objective value compared to heuristic approaches. This small trade-off highlights that C-MICL remains highly competitive while offering formal probabilistic guarantees feasibility. Figure further highlights the computational advantages of our approach, showing that it achieves optimal solutions several orders of magnitude faster than the baseline methods. In summary, method offers compelling combination of computational speedup and theoretical probabilistic guarantees, while preserving comparable objective performance.
5.2 Multi-classification setting
We consider a food basket design problem aimed at minimizing the cost of a basket composed of 25 commodities, subject to nutritional constraints covering 12 nutrients and learned palatability requirement [68]. Palatability, reflecting the appeal or acceptability of the basket to recipients, is inferred from historical data previously deployed baskets and community feedback. Following the formulation in Fajemisin et al. [1], we impose a minimum palatability threshold of 0.5 (consistent with the value proposed in Maragno et al. [7]) ensure that the basket is not only nutritionally adequate and cost-effective, but also socially and culturally appropriate.
We use the dataset of 5,000 food baskets from Maragno et al. [7]. where palatability scores range continuously in [0, 1]. frame this as a classification task, we discretize the scores into four categories (bad, regular, good, and very good), using thresholds at 0.25, 0.5, and 0.75, respectively. Consistent with Maragno et al. [7], we constrain the optimization problem to only include baskets labeled good or very good. As mechanistic oracle exists for palatability, we train regression neural network on all 5,000 samples to approximate this function. The model's predictions are thresholded to produce the categorical ground truth used in experiments. For consistent comparison across MICL, W-MICL, and C-MICL, we benchmark all methods on a subset of 2,500 baskets data points.
We evaluate classification performance using ReLU NNs [9], the only model type readily supported by OMLT for classification tasks [65]. Despite this, our method remains aagess-g,,sttc,, as demonstrated in the regression case study. For MICI and W-MICL (with ensemble sizes P = (5, 10)), we train on all 2,500 data points. For our proposed C-MICU approach, which requires only one trained model, use 2,300 points for training and reserved 200 for conformal calibration. Full details network architecture and training hyperparameters are provided in Appendix D.2.
As shown in Figure 4, only our C-MICL method achieves the desired empirical coverage at 10%, in line with Theorem 4. This highlights the importance of formal statistical guarantees, especially under data constraints and high gigh-dimnnsioall inputs. Figure 5 shows the relative optimality gaps across methods, with differences averaging around 1%. However, none the baselines yield implementable solutions, as only our method satisfies the desired empirical coverage guarantee. Figure 6 shows that C-MICL achieves solution times comparable to solving a single MICL model and significantly faster than the W-MICL heuristics. In both figures, lighter bars indicate methods that do not meet

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1.0
1 a
C-MICL (Ours)
0.8
Feasiblity Rate
W-MICL(10) [7]
W-MICL(10)
Ground-Trutii
[7]
0.6
W-MICL(S), [7]
1-00-
W-MICL(S)
0.4
[7]
MICL
1001
MICL [9]
0.2
[9]
0
10
10
Io+
0.0
4% Optimal Value w.r.t. C-MICL (Ours)
Time Obtain
191
W-MICLISI WIMIUL0))
(Durs)
an Optimal Solution
Figure 4: Empirical ground- truth feasibility rate of optimal solution across MICL methods. Our C-MICL approach (right- most bar) is the only method that consistently satisfies the tar-
Figure 5: Percentage difference in optimal value from NN-based MICL methods compared C- MICL. Difference is around 1% however all baseline approaches (lighter intervals) fail to achieve
Figure 6: Average computa- tional time to find optimal solution from NN-based MICL methods, across 100 runs. C-MICL achieves solution times comparable to single-model MICL, and is significantly faster than the W-MICL heuristics. Error bars represent 95% Cls.
get ground- -tunt feasibility rate 2 90% across 100 runs. Error bars represent 95% Cls.
the desired feasibility guarantee. Results across 100 runs, with 95% Cls.
the target empirical coverage. In summary, method is the only to guarantee ground-truth feasibility, preserves objective performance, and scales efficiently in classification tasks.
Discussion
Our work introduces C-MICL, aggnosic framework for incorporating learned constraints into optimization problems with probabilistic feasibility guarantees. C-MICL supports both regression and classification tasks, leverages conformal prediction to construct uncertainty sets, and encodes these into MIP- MIP-coopatible formulations. It avoids the scalability bottlenecks of ensemble- based methods by requiring training at most two models, achieving substantial computational gains while maintaining comparable high-qualit solutions. Furthermore, since our proposed method separates training from optimization, the size of the resulting optimization formulation is independent of the amount of data used to train the models, unlike previous end-to-end approaches (e.g., Zhao et al. [48]). Similarly, C-MICL can be integrated with existing methodologies aimed at improving the solution quality of standard MICL and W-MICL formulations, such as the enlarged convex hull trust-region approach proposed in Maragno et al. [7] or the trust-region- must-region-iltte framework in Biegler [18]. C-MICL extends naturally to learning objective functions, providing probabilistic guarantees true objective values, or to multiple learned constraints, representing a promising direction for future research on conformal-enhancee optimization.
One limitation of our work, common to general conformal prediction methods, is that when the underlying machine learning model performs poorly, the resulting conformal sets can become too wide and uninformative, potentially rendering the C-MICL problem infeasible. Moreover, although we believe the conditional independence assumption between feasibility and coverage is reasonable in setting, it may fail in cases where the feasible region biased toward data regimes in which conformal coverage does not hold. Nevertheless, assumption of this nature is necessary to handle the inherent dependence between C-MICL feasible solutions and the calibration set Deat. which directly defines the feasible region and would otherwise invalidate standard conformal guarantees. This represents a theoretical compromise for integrating conformal prediction into constrained optimization and enables statistical guarantees that were previously unattainable in MICL approaches.
This work contributes to the integration of machine learning and mathematical optimization by pro- viding a principled framework for embedding data-driven constraints into decision-making processes with statistical guarantees. By ensuring marginal feasibility without access to the true constraint func- tion, C-MICL advances the state of constraint learning, extends predict-then-optimize pipelines, and offers a tractable approach to optimizing over uncertain, learned constraints in real-world applications.
10

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A Machine Learning Models as MIPs
In the following we provide an overview on how to formulate off-the-shelf pre-trained machine learning models mixed-integer programs (MIP) into constraint learning formulations.
A.1 ReLU Neural Networks
We consider feed-forward neural networks composed of fully connected layers with ReLU activations, which can be exactly represented using MIP [9, 10]. Let denote the input variables to the network and y its output variables. All other internal quantities (preactivations, activations, and indicator variables) are distinct auxiliary decision variables introduced solely for encoding the network structure. Therefore, each neuron computes a transformation of the form:
a = max (0, w Ts 8 b)
(7)
where s are the activations from the previous layer, and (w. b) are fixed, trained parameters. The ReLU nonlinearity encoded via the so called big-M -M formulation:
a 2 w 8 s +
S w s b-
-1L
(8)
a S 6U
8 {0,1}
where the activation output, 8 is binary indicator for whether the neuron is active, and L, U tight lower and upper bounds on the pre-activation w T + b. The full network is encoded layer by layer starting from the inputs æ, recursively defining internal arevaativssions and activations, until reaching the output layer
For further details and alternatives MIP reformulations of ReLU neural networks, we refer readers to a recent review by Huchette et al. [69]. While multiple equivalent encodings exist, they yield the same global solution [53], differing only in computational efficiency, and thus do not affect our discussion on ground-truth feasibility.
A.2 Tree Ensembles and Linear-Model Decision Trees
We encode trained tree ensembles (including random forests and gradient-boosted trees) as well as linear-model decision trees using unified MIP formulation based on leaf selection [11, 16, 17J. Let T denote the set of decision trees in the ensemble, and Le the set of leaf nodes tree t T. Each tree partitions the input space into disjoint regions (leaves), each associated with constant prediction value. We introduce binary variables Tı, E {0, 1} indicating whether leaf E C, tree is selected. valid configuration activates exactly one leaf per tree:
Σ Ema=1 VtET FEC
Each internal node in a tree performs a threshold split of the form D;; < oijj where vi a threshold value for input feature z4. We introduce binary variables Way {0, 1} to model these comparisons, where Wi = 1 if, and only if, the condition is satisfied. Monotonicity constraints wi,j S wi,t+ ensure consistency across thresholds, reflecting the ordered structure of decision splits.
Each leaf in tree is reachable only if all the splitting conditions on the path to that leaf are satisfied. For each split node in tree t, let i(s) be the split variable and j(s) the corresponding threshold index. Let Left i.s and Right denote the sets of leaf nodes in the left and right subtrees of s. We enforce:
Σ Tie S (i())
Σ
The -1-00( VIET, E Splits(()
(10)
LELofo
RERight
The final prediction is computed as weighted sum of the selected leaves: the weights are stage- dependent for gradient-booste trees [11], uniform across trees for random forests [16], or equal to linear function of the input features within the selected leaf for modar- decision trees [17].
16

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B Proof of Main Results
B.1 Mondrian Conformal Prediction
Given calibration data {(X1; Vi).....((N,, Yv),. Mondrian conformal prediction a general framework that extends full conformal prediction by enforcing conditional coverage guarantees across a pre-defined set of finite groups defined in terms of both the features variables X and the outcome variable Y [57, 60]. Formally, for group-indicator function g(X,Y) {l,...,K} Mondrian conformal prediction construct conformal sets Co (XN+1) such that
I((Y+11 E (XN+1) ((X++1 YN+1) = k) 2 1 1.
for all groups E [K] = (1,...K)
The core insight of Mondrian conformal prediction lies in computing different conformal quantiles for each group E [K]. Then, for a test point X we generate hypothetical points (X (XN+1,3) for each possible value of the outcome variable y, and compare the conformal score s s(XN+1: V) to
the conformal scores {s(X,, Yi))eee Nexxxe for all calibration data points in the same group as
(1)- That is, the group-specific comparison is performed for E Ig(XN+1.w) {(i E [N] g(X,, K.) = g(Xxx v))}.
This approach yields the following Mondrian conformal sets
C. (x tu = s(XN+1,V) S Ho
(11)
where oiaa is the group-specific quantile
h-a Quantile ((x(X,, Yi) (1 a1 1/
The conditional coverage guarantee is formalized in the following theorem:
Theorem B.1 (Theorem 4.11 in [57]). Assume that (X1, random variables and s a symmetric score function. Then, the Mondrian conformal set Co (X from eq. (11) satisfies the conditional coverage guarantee
YN+1) are exchangeable
P(Yx+1 E Cx((N+1) 1(N+11, YN+1) = k) 21-a
(12)
for all groups k E {I,...K) K} with ( YN+1) = k) > 0, for any user-specified coverage level 10.
Thus, Lemma 3.1 directly applies Theorem B.1 using the group identifier function g(z, v) 1 (y E V). which separates the data according to the ground-truth feasibility condition. Note that the condition ( YN+1)= = k) > from Theorem B. assumes that both feasible and infeasible points can be sampled from the underlying distribution Pxy. which is given by construction of the constraint learning optimization problem.
B.2 Proof of Theorem 4.1
Proof. We first show that Lemma 3.1 and Assumption 4.1 imply that we have appropriate coverage for the feasible points of the C-MICL problem FN = {(x, 2) E X g(x,2) < C, y.).
By the law of total probability,
P(h(x) E (x) | 2) E Fx) =P(()(((z) E H ))P(()))))))))))))))) e (x. (7 ( EN)
+ HHN Fx)
- 1101--//000e/1()2-)===---------- $ ) (x,
=(1 -
+P)(()()))))) & y (z,2) E Fw)]
=(1-a),
where the second equality follows from the conditional independence assumption 4.1
P(((()) E (x) | (z,=) E FN. E )) = P(h()) E (x)( h(x) E e)),
P(h(x) E (x) (z.z) e Fv. & )) = P(h(x) E Cx(x) h(z) & )))
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Moreover, the first inequality follows from the ground ground-truth truth feasibility conformal coverage guarantee in Lemma 3.1 level
min (P(((()) E (a) h(x) P(h()) E Ca)
h(x) ))) 2 -
where the probabilities taken with respect to both the calibration data Deal and the test point.
The previous result says that when a solution (a', ) E FN is feasible, its true function value h(²²) falls within C. (a") with probability at least 1-a. Since Ca(x') is a subset of y for feasible solutions to the C-MICL problem, whenever h(z') is in (x'), must also be in V. Therefore,
P(h(a") V (x',=') E FN) 2 P(M(2)) E C.(2)) (6' Ex) 21-a,
which proves the ground-truth feasibility guarantee.
B.3 Discussion on the Conditional Independence of Feasibility and Coverage Assumption
Assumption 4.1 states that, conditional on ground-truth feasibility (h(z) E y or h(x) $ Y), the event of C-MICL feasibility ((x, 2) E FN = {(xr 2) E X g(x, S 0, C, (2) CV is independent of whether the conformal set contains the true function value (h(x) E C. (z)). To motivate Assumption 4.1 and clarify when it is plausible, we first emphasize that the ground-truth feasibility (GTF) conditional coverage guarantee from Lemma 3.1 can be achieved
fully data-driven way (e.g.. using Mondrian conformal prediction or other label-conditional conformal methods):
((h(2) E Cx()) h(x) E y) 2 1 - P(((z) E Ca(x) h(z) $ y) a
However, in C-MICL we aim to guarantee coverage over the feasible region of the optimization problem FN = { 2) E X g(z, {
IP (h((z) Ca()) Ex)
If the predictive model h(z)) were perfect (i.e., M(x) h(x)) ) then the regions FN and the ground- truth feasible region F would coincide, and the coverage guarantee would transfer directly. However, since we are interested in the more realistic case where T((z)) is imperfect, FN and F differ in a data-dependent way. In this case, since the feasible region FN is implicitly shaped by the calibration set (via C.. (z:)), there is a natural dependency between the feasible solutions of the C-MICL problem and the calibration data, which invalidates standard conformal guarantees relying on exchangeable calibration and test data. Assumption 4.1 precisely seeks to decouple this dependency: it allows us to approximate conformal coverage within Fx by assuming that feasibility does not systematically bias conformal validity, once conditioned on ground-truth feasibility.
To build intuition, consider partitioning Fy into two disjoint subsets: FN n F and FN n.F Then, Assumption 4.1 implics that FN n (respectively Fx n F is not systematically biased towards a region of F (Fe) that is miscalibrated. Mathematically,
P (h(x)) E Ca(a) (a,2) e nF)
IP (h(z)) Ca(x) (z,2) E FN. h(x)) e e))
= PP((()) E Ca(() h(x))())) E))
(h(()) e (z) (z, FN = IP (h(z) Ca(x) (z,z) E FN. h(x)) ))
z P((()) E Ca(x) h(x)) $ ))
These enable us translate the conformal coverage guarantees from the ground-trufe feasible region F to the feasible set FN used in the optimization. Assumption 1.1 is therefore reasonable when the calibration data adequately covers the parts of the input space that intersect the feasible region FN. both within the ground-truth feasible region F and its complement F In this sense, it aligns with standard generalization assumptions that require the training and calibration data to representative of the regions where predictions are deployed. In our experimental settings, we observe good empirical alignment between target and achieved coverage (Appendix E), suggesting that Assumption 4.1 holds reasonably well in practice in realistic data scenarios.
18

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Alternatively, Assumption 4.1 can be approximated using more granular conditional conformal methods, by partitioning the optimization region into finer subregions and enforcing local coverage guarantees within each, which can then be translated to the feasible set FN- However, the assumption may break down if the feasible region FN heavily concentrated areas where the calibration set sparse or systematically miscalibrated.
C Conformal Set MIP Reformulations
The reformulation of the conformal set regression setting straightforward, as the conformal prediction interval is already algebraic in nature and can be directly expressed as:
(a) sV = [M(x) + ύ1-a" -(xx) C [ V
(13)
{ M(x) + Q1 ū(x) S y, Mx) - h1a ((x) 2
(14)
Here, we include only the two non-dominated constraints i.c., those constraints not already implied by the others, to avoid redundancy. Note that these are linear constraints in terms of the estimated Mxx) and ((x), which are assumed to be MIP-representable, and a constant ii-o computed offline during the conformalization procedure.
The classification setting introduces additional modeling complexity, as the inclusion of classes the prediction set must be encoded using mixed-intege constraints. Specifically, the condition
1{-h(x)* I 91-a} E
(15)
represents indicator that is activated (i.e., equals one) if and only if class k is included in the prediction set. To enforce correct classification behavior, we require that at least one desired class k Jodes is predicted (i.e., its indicator is activated), and no undesired class k E ICund is allowed in the prediction set. To model this, we introduce a binary decision variable for each class k E IC:
{
1 if class k is included in the prediction set.
VKEK
(16)
0 otherwise.
Using these variables, the desired classification logic can be encoded via the following mixed-intege integer constraints:
Ca(z) Cy = -Mxx** a A l: E Kunnd
(17)
-h(x)* - -Q1-a S M(1 - ws)
AKEK AkEK
M(r)** k + + + < Mws
Σ wk 21
(18)
kEA
A ls E K
wkk
Here, e is small numerical tolerance (e.g., 10-6)-1-00 to ensure strict inequality, and M is a large positive constant used in the big-M reformulatonn. A practical choice for M the maximum absolute value of the predicted logits observed in the calibration dataset, scaled by safety factor (e.g., 4):
M = 4 max (ax)
(19)
For detailed discussion selecting valid big-M values and their impact computational perfor- mance, we refer the reader to Trespalacios and Grossmann [70].
Note that the conformal prediction sets (x) used in our formulation of C-MICL have predictable,
well-structured forms that are always MiP-representable, making the containment constraint C, (x) S ) tractable regardless of the underlying prediction model.
19

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D Implementation Details
All computations were performed on Linux machine running Ubuntu, equipped with eight Intel@, Xeon@, Gold 6234 CPUs (3.30 GHz) and 1 TB of RAM, utilizing total of eight hardware threads.
D.1 Regression
This case study focuses on the optimal design and operation of a membrane reactor system used for the direct aromatization of methane. This integrated unit enables the conversion of methane into hydrogen and benzene, achieving simultaneous chemical reaction and product separation. By selectively removing hydrogen through membrane, the reactor leverages Le Chatelier's principle to drive the equilibrium forward, resulting in higher methane conversion rates [66].
The optimization problem involves five key decision variables: the inlet volumetric flow rate of methane (Ny), the inlet flow rate of sweep gas (VHe), the operating temperature (7), the tube diameter (de). and the reactor length (L). These variables were sampled uniformly within physically reasonable bounds, as shown below:
V/ ~ U(450, 1500) cm//
UHe ~ U(450, 1500) cm²/h
d, ~ (005, 2.0) cm
L ((10, 100) cm
T' (997.118 1348.12) K
The resulting samples were used as initial conditions for solving the system of ordinary differential equations governing the reactor, enabling the computation of the outlet benzene flow (Fc,H,). The numerical integration was performed using code available in the opyrability repository [71]. To simulate measurement uncertainty, Gaussian noise was added to the computed benzene flows.
Table 1 shows an illustrative subset of the 1,000 data-points generated:
Table 1: Sampled decision variables and simulated benzene outlet flow.
00 (cm²/h)
UHe /h)
T' (K)
de (cm)
L (cm)
FC,H, (mol/h)
1157.75
845.25
1272.93
0.57
30.67
37.45
773.89
1319.69
998.56
0.75
88.10
42.57
1484.77
476.68
1201.21
1.64
77.38
40.70
817.68
536.65
1123.28
0.72
86.07
28.26
1162.73
1262.03
1256.59
1.95
92.29
36.58
All hyperparameters used across the single-model baselines, wrapped approaches, and our proposed methods were kept consistent to ensure fair comparison. For the the Linear-Model Decision Tree, we set the maximum depth to five, the minimum number of samples required to split an internal node ten, and the number of bins used for discretization to forty. For the Random Forest, we used fifteen estimators, maximum depth of five, a minimum samples split of three, and considered sixty percent of the features when looking for the best split. The Gradient Boosting Tree model was configured with fifteen estimators, learning rate of 0.2, a maximum depth of five, a minimum of five samples per split, and sixty percent of the features considered at each split. Lastly, the ReLU Neural Network was set up with two hidden layers of 32 units each, L2 regularization strength of 0.01, and trained for 2000 epochs using Adam optimizer.
All models were cross-validated using 5-fold split of their respective datasets. For the single base models, the complete 1,000 point dataset was used for training and evaluation. In the case of the ensemble methods, bootstrapping the 1,000 data-point dataset was employed to generate the required 500 data-point (half sized) subsets for training following [7]. For our conformal method, split the training data (800 data points) was used, ensuring that the calibration data was kept separate (200 data points). The Table presents the average Mean Squared Error (MSE) across all folds for each of the methods, considering that the variance of the output for the 1,000 data-points is 1.2871.
20

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Table Average MSE across folds for all Methods for reactor model.
Predictive Method
Model
Approach
MSE
RandomForest
MICL
0.1748
ReLU NN
MICL
0.0597
GradientBoosting
MICL
0.1192
LincarModelDT
MICL
0.0634
RandomForest
W-MICL(5)
0.1712
ReLU NN
W-MICL(55
0.070
GrasientBoosting
W-MICL
0.1298
LinearModelDT
W-MICL(5)
0.0594
RandomForest
W-MICL(10)
0.1711
ReLU NN
W-MICL(10)
0.0699
GradientBoosting
W-MICL(10)
0.1239
LinearModeIDTT
W-MICL(10)
0.0568
RandomForest
W-MICL(25) 0.1696
ReLU NN
W-MICL(25)
0.0689
GradientBoosting
W-MICL(25)
0.1217
LinearModelDT
W-MICL(25)
0.0568
RandomForest
W-MICL(50)
0.1665
ReLU NN
W-MICL(50)
0.0685
GrasientBoosting
W-MICL(50)
0.1194
LinearModelDDT
W-MICL(50)
0.0559
RandomForest
C-MICL
0.1847
ReLU NN
C-MICL
0.0841
GradientBoosting
C-MICL
0.1402
LincarModelDT
C-MICL 0.0696
For all uncertainty models ((z). employed ReLU-based Neural Networks with shared architec- ture and regularization setup to maintain consistency across approaches. Specifically, each model used two hidden layers with 32 units each and L2 regularization (weight decay) coefficient of 0.001. The only hyperparameter that varied during cross-vass vooss-vaiiatioon was the number of training epochs, which does not influence the MIP optimization outcomes but affect the quality of the uncertainty estimates. The optimal number of training epochs identified through cross-validation follows: 1000 epochs for the model paired with Gradient Boosting, 3000 epochs for the one used with the ReLU Neural Network (using Adam optimizer), and 2000 epochs for both the Random Forest and Linear-Model Decision Tree variants. Table displays the average MSE for the uncertainty model of each base predictor across all folds.
Table 3: Uncertainty average MSE for each base model.
Base Model
((x)) MSE
GradientBoosting
0.0142
ReLU NN
0.0247
RandomForest
0.0408
LincarModeIDT
0.0225
Tables 4, 5,, and 6 present the sets, parameters and decision variables of the problem, respectively.
Table Sets used in the reactor optimization model.
Symbol Description
Set decision variables
(e.g., I = {v0, v_He, T, dt, L))
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Table 5: Parameters used in the reactor optimization model.
Symbol
Description
Ci
Operational or design cost coefficient associated with variable Z
Table Decision variables the reactor design model.
Symbol
Description
Value of design variable E I, where I = {vo, v_He, т, dt, L} Predicted outlet flow of CgHş from the surrogate model
y
Formulation.
min
Σ q34
(D.la)
(achee
IET
s.t.
10 I
< 150
(D.1b)
Tdt
0.75 S I.0 3.0
(D.1c)
20 S Tv0
I 120
(D.1d)
TL
2.0 S 1.1 . TT
(D.le)
M(x)) y
(D.1f
2 50
(D.lg)
24 ViET
(D.
Explanation Constraints.
(D.la): Minimize the operating cost linear function of the decision variables.
(D.1b): Enforce physical bounds on the tube length- -intmett- ratio.
(D.ic): Maintain suitable gas feed ratio between CH, and He.
(D.id): Control the residence time via the CH, flow and reactor length.
(D.le): Limits methane flow rate based on temperature to ensure safe and stable operation.
(D.1f): Enforces that the surrogate model output y. predicting C& flow, is computed from the design and operation variables
(D.1g): Requires that the predicted CgHHg outlet flow (i.e., product quality) meets exceeds the target value of 50.
(D.1h): Enforces nonnegativity for all design variables.
D.2 Classification
We now examine the food basket optimization problem introduced earlier and grounded in the work of Peters et al. [68]. Our analysis is based on the model developed by Fajemisin et al. [1], which aims to minimize the cost of assembling a basket consisting of 25 different commoditics while satisfying nutritional requirements across 12 key nutrients. An additional constraint is placed on the palatability of the basket, which reflects how acceptable or appealing the food to the target population. In line with the case study by Maragno et al. [7], require minimum palatability score of f 0.5 to ensure that the resulting food baskets are not only affordable and nutritionally adequate but also culturally and socially acceptable. The dataset used in this analysis is publicly available and published by Maragno et al. [7] here.
The palatability scores initially range continuously from 0 to 1. To transform this into classification problem, we discretize these scores into four distinct categories: bad, regular, good, and very good. This discretization is achieved by setting thresholds at 0.25, 0.5, and 0.75, Consequently, the problem
22

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becomes a multi-class classification task, where each food basket is assigned one of these categorical labels based on its palatability score. We restrict the optimization problem include only food baskets that fall within the good and very good categories. This approach is inspired by the work of Maragno et al. [7], where a similar threshold of = 0.5 was used, but applied in a categorical context. Table 7 displays sample of the dataset available.
Table 7: Sample of food basket dataset with commodity quantities and palatability score. Full dataset available here.
Beans
Bulgur
Cheese
Fish
Mcat
CSB
Dates
DSM
Palatability Class
0.7226
0
0
0.5987
0.199
Bad
0.7860
0
0
0
0
0.0419
0
0,3705
0.8049
Very Good
0.4856
0
0
0
0
0
0
0.2696
0.6517
Good
0
0
0.5734
0
0
0
0
0.0025
0.3220
Regular
In this case, we exclusively trained ReLU-based neural networks across all methods (single-model baseline, wrapped approach, and proposed method) ensuring that all hyperparameters remained identical for fair comparison. For the predictors, the network architecture consisted three hidden layers with 64 units each, trained for 500 epochs with weight decay (L2 regularization) of 0.01. Additionally, we trained an oracle model using all available data. This oracle network was configured with five hidden layers of 256 units each, trained for 1000 epochs, also with weight decay of 0.01.
Table 8 displays the average accuracy across validation folds on the data available for each approach.
Table 8: Average accuracy across folds for all methods for the basket model.
Predictive method
Model
Approach
Accuracy [%]
ReLU NN
MICL
84.32
ReLU NN
W-MICL(5)
78.24
ReLU NN
W-MICL(10)
76.88
ReLU NN
C-MICL
83.30
Tables 9, 10, and
present the sets, parameters and decision variables of the problem, respectively.
Table 9: Sets used in the basket model.
Symbol
Description
M
Set of commodities (c.g., M {rice, beans, salt, sugar, ...})
C
Set of nutrients (e.g., L = (protein, iron, calories, ...))
Kddes
Set of desired categories (e.g., Audes (good, very good))
Kuund
Set of undesired categories (e.g., K = [bad, regular))
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Table 10: Parameters used in the basket model.
Symbol
Description
Nutreq,
Nutritional requirement for nutrient E L (grams/person/ddyy)
Nutvalm
Nutrient content of commodity m for nutrient / (grams per gram)
Pm
Procurement cost of commodity m (in $/metric ton)
M
Large positive constant for big-M reformulation.
Table 11: Decision variables in the basket model.
Symbol
Description
Im
Quantity of commodity m E M in the food basket (grams) Palatability level of the food basket
Binary indicator for predicting desired category & Kcdes ((0,1
Formulation.
min
Pm
(D.2a)
{arm MM
meM
s.t.
Σ
Nutvalml Im 2 Nutreq
VlEL L
(D.2b)
mEM
Usalt
(D.2c)
T Fsugar 20
(D.2d)
h(x) = y
(D.2e)
Vk
Uk M(1 wk)
A ke E Kdles k' Kind
(D.2f)
Σ wk 2 1 ke
(D.2g)
Zm 0
WmE M
(D.2h)
Explanation of Constraints.
(D.2a): Objective function minimizing total procurement cost of the food basket.
(D.2b): Nutritional constraints to ensure daily nutrient requirements are met.
(D.2c)(().2)): Fixed amounts of salt and sugar imposed (e.g., due to guidelines).
(D.2e): Palatability computed as a function i((=)) of the selected commodity quantities.
(D.2f): Big-M constraint that activates an indicator if the logit of a desired class k E Kides is higher than the logit of all undesired classes k' E Kuund representing desired prediction. M is calculated using the largest value in magnitude observed in calibration data times a enlarging safety factor (e.g., 4) as M = 4 max; jå (a::) |. For more information on how to calculate valid M values and their impacts optimization we refer the readers to Trespalacios and Grossmann [70].
(D.2g): Enforce least desired class to have larger logit that both undesired classes.
(D.2h): Ensures quantity nonnegativity.
E
Complementary Results
To quantify uncertainty and support the statistical reliability our results, we report 95% confidence intervals (Cls) for all key metrics: empirical feasibility rates, optimization times, relative differences in objective value, and true coverage. All uncertainty comes from 100 problem instances optimization problems.
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For feasibility rates and true coverage plots intervals are computed as the proportion of feasible solutions over meaning that we model each as a Bernoulli random variable and apply the standard emor formula for proportions:
p(1
SEM
where P is the sample mean and is the number of problem instances. The corresponding confidence interval is calculated using the Student's t-distribution:
Cl95% = p - 1 SEM
For optimization times and relative differences in objective values, we compute the sample mean, standard deviation, and standard error of the mean (SEM) for each method across instances. The 95% confidence intervals are then given by:
Cla5% - 11.977
vn
where s the sample standard deviation and n = 100 the number of observations. This method assumes approximate normality of the sample means, which reasonable due to the Central Limit Theorem given the sample size.
All error bars shown in plots correspond to these 95% confidence intervals. The factors of variability captured are due to the random generation optimization instances and the resulting performance metrics across different methods. These intervals are reported directly in the results section and supporting figures to substantiate claims about statistical significance and performance differences.
F.
Regression
The following results summarize performance the reactor case study at 10%, evaluated after solving 100 optimization instances for each method. More specifically, Figure reports the average computational time required to obtain an optimal solution for each method at a 10%, while Figure presents the relative difference optimal objective value between bascline methods and our proposed C-MICL the same confidence level. Lighter bars in both figures denote methods that failed to achieve the target empirical feasibility rate, as established in Figure 1. Additionally, Figure 9 shows the distribution of true constraint values hxx)) for each methods for 10%. Finally, Figure 10 reports the empirical coverage over 1,000 out out-of-sample data points, stratified by deciles of the true output variable For each decile, report the proportion of instances where the true value lies within the predicted interval, demonstrating strong coverage across the output space for all baseline models. Notably, our ReLU NN-based conformal sets empirically satisfy the ground-truth feasibility coverage guarantee from Lemma 1. achieving coverage rates of 89.31 % and 95.38% on ground-truth infeasible and feasible samples, respectively.
We report here the full results for the reactor case study at 5%, based on evaluations conducted over 100 solved optimization instances per method. Specifically, Figure 11 reports the empirical ground-truth feasibility rate achieved by each MICL approach. Figure 12 shows the average computa- tional time needed to solve each method optimal objective value between baseline methods and our C-MICL approach. Furthermore, Figure 14 presents the distribution of true constraint values h(x) for each method at 5%. Finally, Figure 15 reports the empirical coverage the models over 1,000 out-of-sample data points, grouped by deciles of the true output y. Coverage is measured the proportion of points for which the true value of falls within the predicted interval at a target level of a = 5%. Our ReLU NN-based conformal sets demonstrate empirical compliance with the ground-truth feasibility coverage guarantee established in Lemma 3.1, yielding coverage rates of 96.92% for ground-trun -ruth feasible instances and 96, 66.09 for ground-truth infeasible ones.
5%, and Figure 13 displays the relative difference in
25

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Time [s] Obtain an Optimal Solution
RelU NN
107
Gradi Broostng
RandomForest
LinearModel DT
10a
10a
10
MICL 19,11,16,17]
W-MICL(S) [7] W-MICL(10) [7] W-MICL(255 [7]
W-MICL(50) [7]
C-MICL (Ours)
Figure Average computational time obtain an optimal solution for methods
10%. Our
proposed C-MICL approach is comparable to single-model baselines while being orders of magnitude faster than ensemble heuristies using the same underlying base model. Lighter bars indicate methods
that failed to achieve the target empirical feasibility rate.
4% Optimal Value w.r.t. C-MICL (Ours)
H
-10
-20-
GradientBoosting
RandomForest
-25
LinearModelDT
MICL (9,11.16,17]
W-MICL(55 (7)
W-MICL(10) [7]
W-MICL(25) [7]
W-MICL(50) [7]
Figure 8: Relative difference in optimal objective value of baseline methods compared to our proposed
C-MICL at a = 10%. Our approach exhibits a small difference of around 8% compared to the
method that achieved empirical coverage, indicating that C-MICL attains comparable solution quality relative to valid approaches. Lighter bars marked with an "x" denote methods that failed to meet the target empirical feasibility rate.
True Feasibility Constraint Value hox)
0.4
0.3
ReLU NN GradientBoosting RandomForest
0.2
LinearModeIDT
MICL [9,11,16.17] W-MICL(S) [7] W-MICL(10) [7] W-MICL(25) W-MICL(50)
C-MICL (Ours)
Figure 9: Distribution of true constraint values h(z) for each method for 10% The dotted line at 0.5 marks the lower bound imposed in the oracle constraint values below this line correspond to true infeasibilities. C-MICL not only yields the fewest violations but also achieves the smallest violation magnitudes when they occur.
26

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1.0
0.8
Coverage
0.6
0.4
GradientBeosting
0.2
Randomforest
LinearModdl
2
6
10
Decile of True Target
Figure 10: Empirical sat-ol--ammlee coverage across predictive models at a = 10%, evaluated on 1,000 test points and stratified by deciles of the true output Our conformal sets achieve strong target coverage. Crucially, the ground-truth feasibility threshold (y 2 50%) lies within the 9th and 10th deciles, where all methods attain valid empirical coverage, empirically supporting the assumptions of Theorem 4.1.
Ground-Truth Feasibillty Rate
0.6
0.4
Boosting RandomForest
0.2
LinearMade DT
MICL 19,11,16,17
W-MICL/S) [7]
W-MICL(11) [7]
W-MICL(25) [7] W-MICL(50) [7]
C-MICL (Ours)
Figure 11: Empirical ground-truth feasibility rate optimal solutions across MICL methods on 100 optimization problem instances for 5%. Our Conformal MICL approach (rightmost bars) consistently meets the target feasibility rate of 2 95% regardless of the underlying base model. In contrast, existing methods show inconsistent performance and often fall short of the theoretical guarantee.
Time [s] Octain an Optimal Solution
Gradient Boossingg RandomForest LinearMode DT
107
10a
10-1
10-1
MICL 19,11
W-MICL(S) (7] W-MICL(10) [7] W-MICL(25) (7) W-MICL(50)
C-MICL (Ours)
Figure 12: Average computational time to solve 100 optimization instances at = 5%. Our proposed C-MICL method matches the speed of single-model baselines and orders of magnitude faster than
ensemble-hase heuristics using the same base model. Lighter bars denote methods that did not reach the target empirical feasibility threshold.
27

---

A% Optimal Value w.r.t. C.MICL (Ours)
-20-
ReLU NN
-60
GradientBoosting
RandomForest
80
LinearModelDT
MICL [9,11,16,17]
W-MICL(S) [7]
W-MICL(10) (7)
W-MICL(25) [7]
W-MICL((5) [7]
Figure Relative difference in optimal objective value between baseline methods and our proposed
C-MICL 5% across 100 optimization problems. Our approach shows modest 8% difference compared to the method that achieved empirical coverage, demonstrating that C-MICL provides solution quality par with other valid approaches. Lighter bars with "x" indicate methods that did not meet the empirical feasibility target. Five outliers were removed for the W-MICL(50) Gradient Boosted Tree model, though the comparison remains statistically insignificant with without them.
True Feasibility Constraint Value htx)
0.6
0.5
0.4
0.3
ReLU NN GradientBoosting
0.2
RandomForest
L'nearModelDT
MICL [9,11,16,17] W-MICL(S) [7] W-MICL(10) [7] W-MICL(25) [7] W-MICL(55) [7] C-MICL (Ours)
Figure 14: Violin plots true constraint values l(()) each method at a = 5%. The dotted line at 0.5 denotes the lower bound enforced in the oracle constraint where values falling below this line indicate true infeasibilities. At this stricter confidence level, C-MICL maintains the highest reliability, producing the fewest and smallest violations across all methods.
1.0
caverage
0.6
0.4
ReLU NN GradientBoosttnn RandomForest LinearMcdelDT
0.0
Decile Target
Figure 15: Empirical out-of-sample coverage at 5% for all predictive models, evaluated on 1,000 test points and stratified by deciles of the true output y. Our conformal prediction sets achieve the desired coverage level across deciles. Notably, the ground-truth feasibility threshold (y > 50) lies in the 9th 10th deciles, where methods achieve valid empirical coverage, providing empirical support for the assumptions in Theorem 4.1.
28

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1.0
1
C-MICL (Ours)
0.8
Ground-Trutti Feasiblity Rate
W-MICL(10) [7]
W-MICL(10)
[7]
0.6
W-MICI(S). [7]
1-00-
W-MICL(S)
0.4
[7]
MICL
1001
MICL [9]
0.2
[9]
0
10 10
10
0.0
A% Optimal Value w.r.t. C-MICL (Ours)
Time Obtain an Optimal Solution
W-MICLISI WIKKU10))
191
(Durs)
Figure 16: Empirical ground- Figure 17: Relative difference truth feasibility rates across in optimal objective value be- MICL methods on 100 optimiza- tion problem instances at
Figure 18: Average time com- pute an optimal solution for each MICL method 100 optimiza- tion instances at 5%. C- MICL matches the runtime of single-model MICL and outper- forms insemble-basee methods by wide margin. Lighter bars represent approaches that did not achieve the target feasibility level.
tween baseline MICL methods and C-MICL across 100 opti- mization problem instances at = 5%. The differences aver- age about 1%, with all methods achieving statistically similar so- lution quality. However, none of the baselines produce imple- mentable solutions, as indicated by the lighter bars, which repre- sent methods that failed to meet the required empirical coverage guarantee. Only our method sat- isfies this guarantee.
5%. C-MICL (rightmost bar)
is the only method that consis- tently achieves the target feasi- bility threshold of at least 95%, in line with theoretical guaran- tees. All baseline methods fall short of this benchmark.
E.2
Classification
The following three figures present the performance of different MICL methods applied to the food basket design problem under a classification setting, with a = 5%. These figures assess the feasibility, objective value, and computational efficiency of the methods across 100 problem instances, highlighting the strengths of C-MICL in comparison to baseline approaches. Specifically, Figure 16 illustrates the empirical ground-truth feasibility rate, Figure 17 shows the relative difference in optimal objective value, and Figure 18 reports the average computational time required to obtain an optimal solution. Finally, Figures 19 and 20 present the empirical coverage achieved by each model across 1,000 out-of-sample points, grouped by the label y category, for a 10% and 5%, respectively. These figures report the proportion of instances which the true value y falls within the predicted conformal set. Our conformal sets empirically satisfy the ground-truth fcasibility coverage guarantee from Lemma 3.1, achieving coverage rates of 88.22% (feasible) and 19. (infeasible) for 10%, and 93.4 41% % (feasible) and 98. 79% (infcasible) for 5%.
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1.0
0.8
Coverage
0.6
Coverage
0.6
0.4
0.4
0.0
Bad
Regular
Good
Very Good
Bad
Regular
Good
Very Good
True Label
True Label
Figure 19: Empirical out-of-sample coverage,
Figure 20: Evaluation of empirical coverage with ReL.U NNs over 1,000 test samples, grouped by the true label, for target level of at 5%.
evaluated over 1,000 test points and stratified by the true categorical label of y at 10%. Across all categories, the predicted conformal sets cap- ture the true label at the desired coverage level. Notably, for the "Good" and "Very Good" cate- gories, (which correspond to the ground-truth fea-
The model produces conformal prediction sets
that contain the true class label the expected
rate. In particular, the "Good" and "Very Good" categories, those relevant to enforcing feasibility, demonstrate approximate empirical coverage, pro- viding evidence consistent with Theorem 4.1.
sibility region) the models approximately achieve valid empirical coverage, lending support to the assumptions in Theorem 4.
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Reviews:
[{"review_id": "thh6t8nPkW", "reviewer": "Reviewer_Pih6", "summary": "This 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.", "questions": "- 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?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "The authors have sufficiently addressed my concerns about Assumption 4.1.", "rebuttal": "Thank you for the thoughtful review and helpful feedback. Below we address weaknesses and questions. \n\n- W1: We thank the reviewer for highlighting these valuable contributions at the intersection of conformal prediction and optimization. We agree they provide important context and will include them in the revised manuscript. While Johnstone & Cox (2021) construct conformal ellipsoids for robust optimization, and Yeh et al. (2023) learn convex uncertainty sets for robust objectives using differentiable conformal layers, C-MICL focuses on conformalizing feasibility regions rather than robustifying objective values. Kiyani et al. (2023) and Patel et al. (2022) use conformal sets for utility-aware or risk-sensitive decision-making, optimizing value-at-risk or ensuring robust output quality. In contrast, C-MICL targets constraint satisfaction under model misspecification, supporting general nonconvex feasible regions and offering distribution-free feasibility guarantees. This leads to a distinct formulation centered on feasibility-aware decision-making rather than robustifying objectives/actions for downstream optimization problems. We thank the reviewer again for pointing out this additional work at the intersection of conformal prediction and optimization. We will include and discuss these additional references in the related work section in the revised manuscript.\n\n- Q1: We thank the reviewer for this important question. The key insight is that the conformal prediction sets $\\mathcal{C}(x)$ used in C-MICL have predictable, well-structured forms that are always MIP-representable, making the containment constraint $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ tractable regardless of the underlying prediction model.\n\nIn the regression setting, the conformal set $\\mathcal{C}(x)$ is always an interval of the form $\\left[\\hat{h}(x) \\pm \\hat{q}_{1-\\alpha}(x)\\right]$. Therefore, the containment constraint $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ simplifies to checking whether this interval lies entirely within the feasible set $\\mathcal{Y}$. When $\\mathcal{Y} = [\\underline{y}, \\bar{y}]$, this reduces to two linear inequalities, which are directly representable in a MIP.\n\nIn the classification setting, the conformal set $\\mathcal{C}(x)$ is always a finite subset of the label space $\\mathcal{K}$, and can be captured using binary indicator variables $w_k \\in {0,1}$ for each class $k \\in \\mathcal{K}$. The constraint $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ then becomes a set of linear conditions over these binary variables.\n\nSince conformal sets in C-MICL preserve these structured forms across different prediction models and conformity scores (intervals for regression, subsets for classification), the containment constraint remains computationally tractable, with detailed MIP formulations for both settings in Appendix C. We appreciate the opportunity to clarify this point and will emphasize in the revised manuscript the structured nature of conformal sets.\n\n- W2 + Q2: We thank the reviewer for highlighting the importance of Assumption 4.1 and for prompting a more detailed discussion of its role, interpretation, and limitations. We agree that this assumption is central to our theoretical guarantees and warrants careful justification. We believe it is reasonable in our setting and necessary to extend conformal guarantees to constrained optimization problems.\n\nAssumption 4.1 states that, conditional on ground-truth feasibility ($h(x) \\in \\mathcal{Y}$), the event of C-MICL feasibility ($(x, z) \\in \\mathcal{F}_N = \\\\{(x, z) \\in \\mathcal{X}: g(x, z) \\leq 0,\\ \\mathcal{C}(x) \\subseteq \\mathcal{Y}\\\\}$) is independent of whether the conformal set contains the true function value ($h(x) \\in \\mathcal{C}(x)$).\n\nTo motivate Assumption 4.1 and clarify when it is plausible, note first that the ground-truth feasibility (GTF) conditional coverage guarantee from Lemma 3.1 can be achieved in a fully data-driven way (e.g., using Mondrian conformal prediction or other label-conditional conformal methods):\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\in \\mathcal{Y}) \\geq 1-\\alpha$$\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\notin \\mathcal{Y}) \\geq 1-\\alpha$$\n\nHowever, in C-MICL we aim to guarantee coverage over the feasible region of the optimization problem $\\mathcal{F}_N = \\\\{(x, z) \\in \\mathcal{X}: g(x, z) \\leq 0,\\ \\mathcal{C}(x) \\subseteq \\mathcal{Y}\\\\}$, i.e.,\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N) \\geq 1-\\alpha$$\n\nIf the predictive model $\\widehat{h}(x)$ were perfect (i.e., $\\widehat{h}(x) = h(x)$), then the regions $\\mathcal{F}_N$ and the ground-truth feasible region $\\mathcal{F}$ would coincide, and the coverage guarantee would transfer directly. However, since we are interested in the more realistic case where $\\widehat{h}(x)$ is imperfect, $\\mathcal{F}_N$ and $\\mathcal{F}$ differ in a data-dependent way.\n\nIn this case, since the feasible region $\\mathcal{F}_N$ is implicitly shaped by the calibration set (via $\\mathcal{C}(x)$), there is a natural dependency between the feasible solutions of the C-MICL problem and the calibration data, which invalidates standard conformal guarantees relying on i.i.d. calibration and test data.\nAssumption 4.1 precisely seeks to decouple this dependency: it allows us to approximate conformal coverage within $\\mathcal{F}_N$ by assuming that feasibility does not systematically bias conformal validity, once conditioned on ground-truth feasibility.\n\nTo build intuition, consider partitioning $\\mathcal{F}_N$ into two disjoint subsets: $\\mathcal{F}_N \\cap \\mathcal{F}$ and $\\mathcal{F}_N \\cap \\mathcal{F}^c$. Then, Assumption 4.1 implies that $\\mathcal{F}_N \\cap \\mathcal{F}$ (respectively $\\mathcal{F}_N \\cap \\mathcal{F}^c$) is not systematically biased towards a region of $\\mathcal{F}$ ($\\mathcal{F}^c$) that is miscalibrated. Mathematically, \n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N \\cap \\mathcal{F}) = \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N, h(x) \\in \\mathcal{Y}) \\approx \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\in \\mathcal{Y}) $$\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N \\cap \\mathcal{F}^c) = \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N, h(x) \\notin \\mathcal{Y}) \\approx \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\notin \\mathcal{Y}) $$\nThese enable us to translate the conformal coverage guarantees from the ground-truth feasible region $\\mathcal{F}$ to the feasible set $\\mathcal{F}_N$ used in the optimization.\n\nAssumption 4.1 is therefore reasonable when the calibration data adequately covers the parts of the input space that intersect the feasible region $\\mathcal{F}_N$, both within the ground-truth feasible region $\\mathcal{F}$ and its complement $\\mathcal{F}^c$. In this sense, it aligns with standard generalization assumptions that require the training and calibration data to be representative of the regions where predictions are deployed. Alternatively, Assumption 4.1 can be approximated using more granular conditional conformal methods, by partitioning the optimization region into finer subregions and enforcing local coverage guarantees within each, which can then be translated to the feasible set $\\mathcal{F}_N$.\n\nWe will revise the paper to better explain this interpretation and expand the discussion on when the assumption might fail. For instance, the assumption may break down if the feasible region $\\mathcal{F}_N$ is heavily concentrated in areas where the calibration set is sparse or systematically miscalibrated. In our experimental settings, we observe good empirical alignment between target and achieved coverage (Appendix E), suggesting that Assumption 4.1 holds reasonably well in practice in realistic data scenarios.\n\n- Q3: Yes, thank you for raising this point. We agree that our original wording is imprecise.  Assumption 4.1 is meant to apply to the full feasible set of the C-MICL problem, $\\mathcal{F}_N = \\{(x,z) \\in \\mathcal{X}: g(x,z)\\leq 0, \\mathcal{C}(x) \\subseteq \\mathcal{Y}\\}$, rather than only to the containment condition $ \\mathcal{C}(x) \\subseteq \\mathcal{Y}$. We will revise the statement to clarify this and more accurately state the meaning of our assumption as used in the proof. Thanks again for pointing this out.\n\n- W3 + Q4: We thank the reviewer for this important question. However, since $\\mathcal{C}(x)$ is defined using $\\widehat{h}(x)$ rather than the true function $h(x)$, the feasible regions defined by the oracle constraint $h(x) \\in \\mathcal{Y}$ and the conformal set containment $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ differ in no clear way. In particular, neither region necessarily contains the other, and therefore one is not more conservative nor yields larger/smaller optimal values in general.\n\nOn the other hand, relative to the learned function $\\widehat{h}(x)$, our conformal constraint is indeed conservative since $\\widehat{h}(x) \\in \\mathcal{C}(x)$ by construction, making the containment constraint $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$ more strict than $\\widehat{h}(x) \\in \\mathcal{Y}$. The latter case corresponds exactly to the MICL approach where point predictions are used directly in the optimization problem without uncertainty quantification. We compare C-MICL to this naive method in Figures 2 and 5, showing optimal values 15\\% and 1\\% smaller for our regression and classification settings, respectively. Crucially, while MICL achieves better optimal values, it systematically violates ground-truth feasibility constraints as shown in Figures 1 and 4. Our method provides probabilistic ground-truth feasibility guarantees while achieving comparable optimal values.\n\nNevertheless, we agree that comparing to the ground truth problem's optimal objective value (i.e., solving the optimization problem using the true constraint $h(x) \\subseteq \\mathcal{Y}$) is a relevant baseline. We will add this to the updated version and discuss the corresponding results."}, {"review_id": "eASMLpnCan", "reviewer": "Reviewer_fgyD", "summary": "The 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.", "questions": "- 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?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "After reading through other referees' comments and the authors' responses, I have decided to maintain my score.", "rebuttal": "Thank you for the thoughtful review and helpful feedback. Below we address weaknesses and questions. \n\n- W1: We thank the reviewer for highlighting the importance of Assumption 4.1. We agree that this assumption is central to our theoretical guarantees and warrants careful justification. \n\nIntuitively, Assumption 4.1 is plaussible when the calibration data adequately covers the parts of the input space that intersect the feasible region $\\mathcal{F}_N$, both within the ground-truth feasible region $\\mathcal{F}$ and its complement $\\mathcal{F}^c$. In this sense, it aligns with standard generalization assumptions that require the training and calibration data to be representative of the regions where predictions are deployed. Alternatively, Assumption 4.1 can be approximated in a data-driven way using more granular conditional conformal methods, by partitioning the optimization region into finer subregions and enforcing local coverage guarantees within each, which can then be translated to the feasible set $\\mathcal{F}_N$.\n\nWe will revise the paper to better explain this interpretation and expand the discussion on when the assumption might fail. For instance, the assumption may break down if the feasible region $\\mathcal{F}_N$ is heavily concentrated in areas where the calibration set is sparse or systematically miscalibrated. In our experimental settings, we observe good empirical alignment between target and achieved coverage (Appendix E), suggesting that Assumption 4.1 holds reasonably well in practice in realistic data scenarios.\n\n- W2: The secondary regression model $\\hat{u}(x)$ introduced in Section 4.1 is used to quantify the uncertainty around the predictive model $\\hat{h}(x)$. This secondary model is essential for constructing adaptive conformal sets whose size depends on the input $x$. As is standard in the conformal prediction literature, such adaptive sets allow us to maintain valid coverage while tailoring the prediction interval to the local uncertainty of the model (see Angelopoulos et al [52] and Papadopoulos et al. [56]). The use of a separate model for estimating uncertainty is a common and well-established practice in conformal regression, and we will revise the manuscript to better introduce and motivate the role of $\\hat{u}(x)$ when it is first presented.\n\n- W3: Thank you for raising this valid and insightful concern about violation magnitude. We evaluated the magnitude of feasibility violantions to check whether C-MICL produces grossly infeasible solutions. Specifically, we inspected the actual values of the constraint violations and confirmed that, even when violations occurred, they were relatively minor. For example, in the NN classification case with $\\alpha = 10\\%$, which was the scenario with the highest number of violations for C-MICL (11 out of 100), the true values were: [0.395, 0.424, 0.455, 0.462, 0.479, 0.482, 0.487, 0.489, 0.489, 0.493, 0.495], under a feasibility threshold of 0.5. These values show that C-MICL tends to stay close to the feasible region even when feasibility is not strictly satisfied. We appreciate the suggestion and will incorporate violin plots to the Appendix in the revised manuscript to visualize the distribution of constraint violations across methods and scenarios.\n\n- Q1: The C-MICL approach remains applicable when the feasible set $\\mathcal{Y}$ is more complicated, for instance, involving integer variables and constraints. While feasibility checking may appear more challenging when $\\mathcal{Y}$ is more complex, the modeling effort lies in formulating the feasible set $\\mathcal{Y}$ as a MIP, which is inherent to any constraint-learning method that must verify feasibility with respect to $\\mathcal{Y}$, not specific to our proposed conformal approach. These MIP-representable constraint sets include finite unions of intervals, discrete sets, and sets defined by (non)linear inequalities, piecewise (non)linear constraints, and logical operations (e.g, disjunctions), capturing a wide class of practical applications in operations research and optimization (see \"50 Years of Mixed-Integer Nonlinear and Disjunctive Programming\" by Kronqvist et al.).\n\nIn such cases, C-MICL still enforces feasibility by verifying that the conformal prediction set lies entirely within the feasible set, i.e., $\\mathcal{C}(x) \\subseteq \\mathcal{Y}$. This condition remains consistent with our formulations  in Equations (4) and (6) for regression and classification, respectively.  We appreciate the opportunity to discuss this point and will clarify the generality of our method subject to $\\mathcal{Y}$ being MIP-representable in the revised manuscript.\n\n- Q2: Thank you for this interesting question. When the $g$ function involves the outcome variable $y$, the conformalization procedure becomes more complex. Since we have a conformal prediction set $\\mathcal{C}(x)$ for $y = \\widehat{h}(x)$, evaluating $g$ requires considering all possible values in this set, yielding a corresponding prediction set $\\mathcal{G}(x,z) = \\\\{g(x, \\widehat{y}, z) : \\widehat{y} \\in \\mathcal{C}(x)\\\\}$. The constraint $g(x,z) \\leq 0$ in the MICL formulation becomes $\\sup \\mathcal{G}(x,z) \\leq 0$, which remains MIP-representable and can be incorporated in our framework.\n\nHowever, extending the coverage guarantees from $\\mathcal{C}(x)$ to the prediction set $\\mathcal{G}(x,z)$ is non-trivial for arbitrary functions $g$. The challenge lies in preserving the coverage properties when the conformal sets are transformed through potentially complex nonlinear functions. While this extension is theoretically possible under certain regularity conditions on $g$, it requires careful analysis of how prediction uncertainties propagate through the constraint function. We agree this is a promising direction for future research and will add this discussion to the updated manuscript.\n\n- Q3: Thank you for this valuable observation. You are correct that our framework requires the predictive model to be MIP‑representable. When we describe the approach as \"model‑agnostic\", we mean that it can accommodate any predictive model for which a MIP formulation exists. In particular, many widely used machine learning models can be formulated as mixed-integer programs  in practice. Examples include linear, polynomial, and symbolic regression models (Wilson et al.), support vector machines (Maragno et al. [7]), and tree ensembles (Mistry et al. [11]). Moreover, several neural network architectures also admit MIP representations, including feedforward networks (Fischetti et al. [8]), graph neural networks (see Hojny et al.), and Kolmogorov–Arnold Networks (Karia et al.). Thus, although we impose the MIP-representability requirement, the framework remains broadly applicable in real‑world settings. We note that this distinction is noted in Remark 3.1 on page 4. Nonetheless, we are happy to clarify the intended meaning of “model‑agnostic” in the revised manuscript.\n\n- Q4: The error bars correspond to 95\\% confidence intervals for: (i) estimated ground-truth feasibility rates in Figures 1 and 4, (ii) average relative differences in objective values in Figures 2 and 5, and (iii) average computational times in Figures 3 and 6. All estimates are computed over 100 randomly generated optimization instances. Specific formulas for these confidence intervals are provided in Appendix E, and we will add this clarification to the figure captions and when discussing the plots in the text.\n\nRegarding the reported ground-truth feasibility of the C-MICL method, the observed variations around the target coverage level stem from finite sample effects rather than theoretical violations. Our estimates use 100 independent optimization instances, introducing natural estimation uncertainty. Crucially, C-MICL contains the nominal level within the confidence intervals across all experiments, indicating no statistically significant deviation (miscoverage) from theoretical guarantees.\n\n- Q5: As correctly noted, our claim of efficiency primarily refers to the fact that C-MICL requires training at most two predictive models, in contrasts to conformal ensemble-based methods which often involve training many models, leading to optimization times that are orders of magnitude longer.\n\nWith respect to scalability in dataset size, we clarify that our claim refers specifically to the  optimization phase. Existing methods, such as that of Zhao et al. [47], embed the entire training dataset as explicit constraints in the optimization model. This results in a formulation whose size grows linearly with the number of training samples, substantially affecting scalability and tractability of the optimization problem. In contrast, our framework trains predictive models offline, and once trained, the optimization problem is independent of the training dataset size. This means that increasing the size of the training data does not impact the tractability or runtime of the optimization phase. For this reason, we did not test different dataset sizes to assess optimization time, as it remains fixed once the models are trained. We appreciate the reviewer’s comment and will revise the manuscript to clarify this distinction more explicitly.\n\n- Q6: Thank you for the thoughtful question. We chose the calibration split ratios to ensure approximately 200 data points in the conformal calibration set, following common practice in the literature, which typically recommends using between 100 and 500 points (see Angelopolous et al. [52]). Naturally, there is a trade-off involved: allocating more data to the calibration set improves the accuracy of the conformal quantile estimation, while reducing the amount of training data may affect the quality of the prediction model. Our choice aims to reflect realistic data limitations while demonstrating that the C-MICL approach remains effective in such settings. We will clarify this rationale and discuss the trade-off more explicitly in the revised manuscript."}, {"review_id": "nJFiJP3Wqi", "reviewer": "Reviewer_zD5K", "summary": "This 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.", "questions": "1.\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?", "limitations": "The 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.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "The paper proposes a novel conformal mixed-integer constraint learning. During the author-reviewer discussion, the authors provided detailed response, which addressed my concerns. So I will maintain my score that is inclined to acceptance.", "rebuttal": "Thank you for the thoughtful review and helpful feedback. Below we address weaknesses and questions. \n\n- W1: We thank the reviewer for highlighting the importance of Assumption 4.1 and for prompting a more detailed discussion of its role, interpretation, and limitations. We agree that this assumption is central to our theoretical guarantees and warrants careful justification. We believe it is reasonable in our setting and necessary to extend conformal guarantees to constrained optimization problems.\n\nAssumption 4.1 states that, conditional on ground-truth feasibility ($h(x) \\in \\mathcal{Y}$), the event of C-MICL feasibility ($(x, z) \\in \\mathcal{F}_N = \\\\{(x, z) \\in \\mathcal{X}: g(x, z) \\leq 0,\\ \\mathcal{C}(x) \\subseteq \\mathcal{Y}\\\\}$) is independent of whether the conformal set contains the true function value ($h(x) \\in \\mathcal{C}(x)$).\n\nTo motivate Assumption 4.1 and clarify when it is plausible, note first that the ground-truth feasibility (GTF) conditional coverage guarantee from Lemma 3.1 can be achieved in a fully data-driven way (e.g., using Mondrian conformal prediction or other label-conditional conformal methods):\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\in \\mathcal{Y}) \\geq 1-\\alpha$$\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\notin \\mathcal{Y}) \\geq 1-\\alpha$$\n\nHowever, in C-MICL we aim to guarantee coverage over the feasible region of the optimization problem $\\mathcal{F}_N = \\\\{(x, z) \\in \\mathcal{X}: g(x, z) \\leq 0,\\ \\mathcal{C}(x) \\subseteq \\mathcal{Y}\\\\}$, i.e.,\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N) \\geq 1-\\alpha$$\n\nIf the predictive model $\\widehat{h}(x)$ were perfect (i.e., $\\widehat{h}(x) = h(x)$), then the regions $\\mathcal{F}_N$ and the ground-truth feasible region $\\mathcal{F}$ would coincide, and the coverage guarantee would transfer directly. However, since we are interested in the more realistic case where $\\widehat{h}(x)$ is imperfect, $\\mathcal{F}_N$ and $\\mathcal{F}$ differ in a data-dependent way.\n\nIn this case, since the feasible region $\\mathcal{F}_N$ is implicitly shaped by the calibration set (via $\\mathcal{C}(x)$), there is a natural dependency between the feasible solutions of the C-MICL problem and the calibration data, which invalidates standard conformal guarantees relying on i.i.d. calibration and test data.\nAssumption 4.1 precisely seeks to decouple this dependency: it allows us to approximate conformal coverage within $\\mathcal{F}_N$ by assuming that feasibility does not systematically bias conformal validity, once conditioned on ground-truth feasibility.\n\nTo build intuition, consider partitioning $\\mathcal{F}_N$ into two disjoint subsets: $\\mathcal{F}_N \\cap \\mathcal{F}$ and $\\mathcal{F}_N \\cap \\mathcal{F}^c$. Then, Assumption 4.1 implies that $\\mathcal{F}_N \\cap \\mathcal{F}$ (respectively $\\mathcal{F}_N \\cap \\mathcal{F}^c$) is not systematically biased towards a region of $\\mathcal{F}$ ($\\mathcal{F}^c$) that is miscalibrated. Mathematically, \n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N \\cap \\mathcal{F}) = \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N, h(x) \\in \\mathcal{Y}) \\approx \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\in \\mathcal{Y}) $$\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N \\cap \\mathcal{F}^c) = \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N, h(x) \\notin \\mathcal{Y}) \\approx \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\notin \\mathcal{Y}) $$\nThese enable us to translate the conformal coverage guarantees from the ground-truth feasible region $\\mathcal{F}$ to the feasible set $\\mathcal{F}_N$ used in the optimization.\n\nAssumption 4.1 is therefore reasonable when the calibration data adequately covers the parts of the input space that intersect the feasible region $\\mathcal{F}_N$, both within the ground-truth feasible region $\\mathcal{F}$ and its complement $\\mathcal{F}^c$. In this sense, it aligns with standard generalization assumptions that require the training and calibration data to be representative of the regions where predictions are deployed. Alternatively, Assumption 4.1 can be approximated using more granular conditional conformal methods, by partitioning the optimization region into finer subregions and enforcing local coverage guarantees within each, which can then be translated to the feasible set $\\mathcal{F}_N$.\n\nWe will revise the paper to better explain this interpretation and expand the discussion on when the assumption might fail. For instance, the assumption may break down if the feasible region $\\mathcal{F}_N$ is heavily concentrated in areas where the calibration set is sparse or systematically miscalibrated. In our experimental settings, we observe good empirical alignment between target and achieved coverage (Appendix E), suggesting that Assumption 4.1 holds reasonably well in practice in realistic data scenarios.\n\n\n- W2 + Q1: We thank the reviewer for raising this important question and for the opportunity to elaborate on this limitation of the C-MICL framework. As noted in the manuscript,  the optimization problem may become infeasible when the conformal sets are overly conservative (e.g., too wide or trivial), which can  occur due to a poorly performing base predictor or high intrinsic noise. In practice, this issue can be diagnosed by evaluating the predictive performance of $\\widehat{h}(x)$, analyzing the distribution of conformal scores on calibration data, and inspecting the  conformal sets on a hold-out test set. \n\nA natural way to mitigate this issue is to improve the underlying predictive model using standard machine learning techniques or to refine the choice of conformal score functions to better capture uncertainty. However, a less obvious but effective approach is to reduce the statistical confidence level used to construct $\\mathcal{C}(x)$. This leads to smaller conformal sets and can restore feasibility, while still providing explicit and interpretable probabilistic guarantees at a potentially lower coverage level.\n\nIn our experiments, we find that with reasonably well-trained models the conformal sets remain informative and the C-MICL problem is typically feasible in practice. Finally, we also note that this limitation is not unique to C-MICL. For instance, MICL and W-MICL may be more sensitive to poor model quality, as they rely solely on point predictions without accounting for uncertainty, which can result in solutions that are not practically implementable. We will clarify this point further in the revised version and thank the reviewer again for highlighting this issue.\n\n\n- Q2: We appreciate the reviewer’s attention to the scope of our C-MICL method. As correctly noted, our theoretical results do not assume linearity in the underlying optimization problems and extend naturally to mixed-integer nonlinear programs. We focused on MILPs in our experiments to ensure global optimality across all methods (MICL, W-MICL, C-MICL) tractably, allowing for fair comparisons and avoiding confounding due to local optima. Our extensive evaluation setup involved solving each method-model pair across hundreds of independent instances, resulting in several hundred globally solved problems per approach. This level of evaluation is considerably more tractable in the MILP setting.\n\nThat said, our C-MICL approach applies directly to MINLPs, and our theory continues to provide probabilistic ground-truth feasibility guarantees. The primary challenge in extending the experimental evaluation to MINLPs is the significant increase in computational effort required to solve these problems to global optimality. We will include this discussion in the updated version of the article, along with a discussion of the specific computational challenges that arise when applying C-MICL to MINLPs."}, {"review_id": "A66ykV7HTo", "reviewer": "Reviewer_o8nC", "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.", "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", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n-  The paper is well-written and well-organized.\n-  The authors clearly define the problem, and the procedure to implement the proposed framework.\n-  The authors validate their approach on both synthetic dataset and real dataset with real-world applications.\n\n\nWeakness:\n-  [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.", "was_revised": true, "final_justification": "The paper is interesting and the authors solve my concerns. I've changed my score accordingly.", "rebuttal": "Thank you for the thoughtful review and helpful feedback. Below we address weaknesses and questions. \n\n- W1: We thank the reviewer for highlighting the importance of Assumption 4.1 and for prompting a more detailed discussion of its role, interpretation, and limitations. We agree that this assumption is central to our theoretical guarantees and warrants careful justification. We believe it is reasonable in our setting and necessary to extend conformal guarantees to constrained optimization problems.\n\nAssumption 4.1 states that, conditional on ground-truth feasibility ($h(x) \\in \\mathcal{Y}$), the event of C-MICL feasibility ($(x, z) \\in \\mathcal{F}_N = \\\\{(x, z) \\in \\mathcal{X}: g(x, z) \\leq 0,\\ \\mathcal{C}(x) \\subseteq \\mathcal{Y}\\\\}$) is independent of whether the conformal set contains the true function value ($h(x) \\in \\mathcal{C}(x)$).\n\nTo motivate Assumption 4.1 and clarify when it is plausible, note first that the ground-truth feasibility (GTF) conditional coverage guarantee from Lemma 3.1 can be achieved in a fully data-driven way (e.g., using Mondrian conformal prediction or other label-conditional conformal methods):\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\in \\mathcal{Y}) \\geq 1-\\alpha$$\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\notin \\mathcal{Y}) \\geq 1-\\alpha$$\n\nHowever, in C-MICL we aim to guarantee coverage over the feasible region of the optimization problem $\\mathcal{F}_N = \\\\{(x, z) \\in \\mathcal{X}: g(x, z) \\leq 0,\\ \\mathcal{C}(x) \\subseteq \\mathcal{Y}\\\\}$, i.e.,\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N) \\geq 1-\\alpha$$\n\nIf the predictive model $\\widehat{h}(x)$ were perfect (i.e., $\\widehat{h}(x) = h(x)$), then the regions $\\mathcal{F}_N$ and the ground-truth feasible region $\\mathcal{F}$ would coincide, and the coverage guarantee would transfer directly. However, since we are interested in the more realistic case where $\\widehat{h}(x)$ is imperfect, $\\mathcal{F}_N$ and $\\mathcal{F}$ differ in a data-dependent way.\n\nIn this case, since the feasible region $\\mathcal{F}_N$ is implicitly shaped by the calibration set (via $\\mathcal{C}(x)$), there is a natural dependency between the feasible solutions of the C-MICL problem and the calibration data, which invalidates standard conformal guarantees relying on i.i.d. calibration and test data.\nAssumption 4.1 precisely seeks to decouple this dependency: it allows us to approximate conformal coverage within $\\mathcal{F}_N$ by assuming that feasibility does not systematically bias conformal validity, once conditioned on ground-truth feasibility.\n\nTo build intuition, consider partitioning $\\mathcal{F}_N$ into two disjoint subsets: $\\mathcal{F}_N \\cap \\mathcal{F}$ and $\\mathcal{F}_N \\cap \\mathcal{F}^c$. Then, Assumption 4.1 implies that $\\mathcal{F}_N \\cap \\mathcal{F}$ (respectively $\\mathcal{F}_N \\cap \\mathcal{F}^c$) is not systematically biased towards a region of $\\mathcal{F}$ ($\\mathcal{F}^c$) that is miscalibrated. Mathematically, \n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N \\cap \\mathcal{F}) = \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N, h(x) \\in \\mathcal{Y}) \\approx \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\in \\mathcal{Y}) $$\n$$\\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N \\cap \\mathcal{F}^c) = \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid (x, z) \\in \\mathcal{F}_N, h(x) \\notin \\mathcal{Y}) \\approx \\mathbb{P}(h(x) \\in \\mathcal{C}(x) \\mid h(x) \\notin \\mathcal{Y}) $$\nThese enable us to translate the conformal coverage guarantees from the ground-truth feasible region $\\mathcal{F}$ to the feasible set $\\mathcal{F}_N$ used in the optimization.\n\nAssumption 4.1 is therefore reasonable when the calibration data adequately covers the parts of the input space that intersect the feasible region $\\mathcal{F}_N$, both within the ground-truth feasible region $\\mathcal{F}$ and its complement $\\mathcal{F}^c$. In this sense, it aligns with standard generalization assumptions that require the training and calibration data to be representative of the regions where predictions are deployed. Alternatively, Assumption 4.1 can be approximated using more granular conditional conformal methods, by partitioning the optimization region into finer subregions and enforcing local coverage guarantees within each, which can then be translated to the feasible set $\\mathcal{F}_N$.\n\nWe will revise the paper to better explain this interpretation and expand the discussion on when the assumption might fail. For instance, the assumption may break down if the feasible region $\\mathcal{F}_N$ is heavily concentrated in areas where the calibration set is sparse or systematically miscalibrated. In our experimental settings, we observe good empirical alignment between target and achieved coverage (Appendix E), suggesting that Assumption 4.1 holds reasonably well in practice in realistic data scenarios.\n\n\n- Q1: Thanks for the opportunity to provide more details on the influence of the specific conformal scores used in our C-MICL framework. In the current work, conformal score functions were selected to maintain general applicability across a wide range of regression and classification models, in line with the model-agnostic nature of the proposed C-MICL framework. Following standard practice in the conformal prediction literature, we employed score functions that are symmetric and proportional to the model's estimated uncertainty (see Angelopoulos et al. [52]). By aligning with widely accepted strategies in the literature, we aimed to provide a robust yet general approach that performs well across various learning settings without relying on model-specific scores.\n\nSpecifically, in our experiments  we use a residual-based score function for the regression setting, as it relies solely on point predictions and can therefore be applied uniformly across all regression models. For classification, the score function was chosen to preserve linearity in the decision space, enabling efficient integration into the underlying optimization problem. \n\nWe agree that exploring the effect of different valid conformal scores and providing guidance for their selection is a valuable direction for future work. There is already a substantial body of literature that investigates alternative score functions and their impact on the resulting conformal sets, which we will add to the revised manuscript. These works highlight the importance of constructing informative conformal sets to ensure both the feasibility and the quality of the downstream optimization problems. We appreciate the opportunity to discuss this further and will incorporate this perspective in the updated manuscript."}]

Author rebuttal:

Meta-review: I would like to thank the authors for submitting an interesting paper on learning constraints from data using a new framework that they call Conformal Mixed-Integer Constraint Learning (C-MICL).  The reviewers are unanimously positive on the paper,  advocating that it provides a new approach for constraint learning from data with probabilistic feasibility guarantees. The paper is well written and provides detailed convincing experiments. Reviewers raised various questions that were overall comprehensively and satisfactorily addressed in the rebuttal. 

I would like to recommend acceptance of the paper for NeurIPS, encouraging the authors to include brief summaries of the key discussions with reviewers -- especially additional scholarship putting the paper in the context of prior work, and discussing the conditional independence assumption. 

Thank you, 
AC.
Decision: Accept (spotlight)