Inverse Mixed-Integer Programming: Learning Constraints then Objective Functions

A data-driven inverse optimization problem (DDIOP) is the problem of estimating the objective-function parameters (weights) that explain observed optimal-solution data, and it arises in many applications, including integer linear programming (ILP). It is known that, by applying gradient-based optimization methods to the suboptimality loss, the inverse optimization of ILPs can be solved exactly within finitely many oracle iterations, and that the required number of iterations is bounded as $T=O(1/γ(\ell_{\mathrm{sub}})^2)$ in terms of a problem-dependent geometric constant $γ(\ell_{\mathrm{sub}})$. However, no means of bounding $γ(\ell_{\mathrm{sub}})$ from below as a function of the problem size has been available, and hence the number of iterations could not be given as an explicit function of the problem size. We therefore give, when the forward problem is an integer linear program (ILP), the number of iterations sufficient for projected subgradient descent applied to the suboptimality loss to achieve exact consistency with the observed data, as a fully explicit function of the number of samples, the dimension of the features, the ranges of the features, and the structure of the constraint coefficient matrix, up to polynomial factors in the basic constants (the diameter of the weight set, the step-size parameter, and the Lipschitz constant of the suboptimality loss).

Production planning in the manufacturing industry often relies on the use of optimization models, but defining an appropriate objective function can be a challenge. In practice, planners must balance competing goals, manage uncertainty, and account for qualitative business preferences that are difficult to quantify. As a result, many optimization models fail to match expert behavior, limiting trust and adoption. In this work, we propose a data-driven inverse optimization framework to infer the objective function implicitly captured in expert planners' decisions. We formulate the production planning problem as a mixed-integer linear program, where the unknown objective function is represented as a weighted sum of hypothesized cost terms. A suboptimality-loss-based inverse optimization method is then applied to learn the objective weights from historical production plans. The proposed approach is applied to a real industrial case provided by Dow, where the inferred weights reveal that avoiding inventory shortages and maintaining consistent cycle lengths dominate the planners' decision-making. Time- and product-dependent extensions further improve predictive accuracy and uncover evolving priorities. Expert interviews confirm the practical validity of these insights. Overall, this study shows that inverse optimization can transform tacit human expertise into interpretable models, enabling more accurate and trusted decision-support tools for complex industrial systems.

Lagrangian Relaxation (LR) is a powerful technique for solving large-scale Mixed Integer Linear Programming (MILP), particularly those with decomposable structures, such as vehicle routing or unit commitment problems. By relaxing the coupling constraints, LR enables parallel subproblem solving and often yields tighter dual bounds than standard linear programming relaxations, which is crucial for efficient branch-and-bound pruning. While recent empirical work has shown promising results using machine learning to predict these multipliers, a theoretical understanding of such methods remains an open question. In this work, we bridge this gap by analyzing the problem of learning LR through the lens of Data-driven Algorithm Design, i.e., a statistical learning problem over a distribution of problem instances. Our contributions are as follows: first, we derive a generalization bound of $\mathcal{O}(s^{1.5}/\sqrt{N})$ for the learned multipliers, where $s$ is the number of coupling constraints and $N$ is the sample size. Second, we provide a minimax lower-bound of $Ω(s/\sqrt{N})$, proving that a linear dependency is unavoidable. Third, we constructively close this theoretical gap by proving that Stochastic Gradient Ascent (SGA) with averaging achieves the minimax optimal rate $Θ(s/\sqrt{N})$. Finally, we extend our framework to the learning-to-warm-start setting, proving that it achieves a fast, minimax-optimal rate of $Θ(s/N)$ and establishing a theoretical advantage over direct multiplier prediction.