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.

In this work, we propose a cutting plane algorithm to improve optimization models that are originally formulated as convex generalized disjunctive programs. Generalized disjunctive programs are traditionally reformulated as mixed-integer nonlinear programming (MINLP) problems using either the big M (BM) or the hull reformulation (HR). The former yields a smaller MILP/MINLP problem, whereas the latter yields a tighter one. The HR can be further strengthened by using the concept of basic step from disjunctive programming. The proposed algorithm uses the strengthened formulation to derive cuts for the big-M formulation, generating a stronger formulation with small growth in problem size. We test the algorithm with several instances. The results show that the algorithm improves generalized disjunctive programming convex models, in the sense of providing formulations with stronger continuous relaxations than the BM formulation, with few additional constraints. In general, the algorithm also leads to a reduction in the solution time of the problems.

In this work, we propose an algorithmic approach to improve mixed-integer models that are originally formulated as convex generalized disjunctive programs (GDPs). The algorithm seeks to obtain an improved continuous relaxation of the mixed-integer linear and mixed-integer nonlinear programming (MILP/MINLP) model reformulation of the GDP while limiting the growth in the problem size. There are three main stages that form the basis of the algorithm. The first one is a presolve, consequence of the logic nature of GDP, which allows us to reduce the problem size, find good relaxation bounds, and identify properties that help us determine where to apply a basic step. The second stage is the iterative application of basic steps, selecting where to apply them and monitoring the improvement of the formulation. Finally, we use a hybrid reformulation of GDP that seeks to exploit both of the advantages attributed to the two common GDP-to-MILP/MINLP transformations, the Big-M, and the Hull reformulation. We illustrate the application of this algorithm with several examples. The results show the improvement in the problem formulations by generating models with improved relaxed solutions and relatively small growth of continuous variables and constraints. The algorithm generally leads to reduction in the solution times.

The goal of this paper is to develop models and methods that use complementary strengths of Mixed Integer Linear Programming (MILP) and Constraint Programming (CP) techniques to solve problems that are otherwise intractable if solved using either of the two methods. The class of problems considered in this paper have the characteristic that only a subset of the binary variables have non-zero objective function coefficients if modeled as an MILP. This class of problems is formulated as a hybrid MILP/CP model that involves some of the MILP constraints, a reduced set of the CP constraints, and equivalence relations between the MILP and the CP variables. An MILP/CP based decomposition method and an LP/CP-based branch-and-bound algorithm are proposed to solve these hybrid models. Both these algorithms rely on the same relaxed MILP and feasibility CP problems. An application example is considered in which the least-cost schedule has to be derived for processing a set of orders with release and due dates using a set of dissimilar parallel machines. It is shown that this problem can be modeled as an MILP, a CP, a combined MILP-CP OPL model (Van Hentenryck 1999), and a hybrid MILP/CP model. The computational performance of these models for several sets shows that the hybrid MILP/CP model can achieve two to three orders of magnitude reduction in CPU time.

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