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Inverse Mixed-Integer Programming: Learning Constraints then Objective Functions

Akira Kitaoka

TL;DR

This work addresses learning both the constraint set and the objective in MILP from observed decisions by proposing a two-stage inverse optimization framework that first learns constraint parameters and then objective weights. The forward problem is formulated as a MILP with a linear-combination objective and constraint maps defined by lattice-homomorphisms, enabling a tractable Solve_IOP learning procedure. The authors establish finite-data solvability, extend statistical learning theory to pseudometric spaces, and derive generalization bounds for inverse optimization, plus an empirical demonstration on a single-machine scheduling ILP with up to 100 decision variables achieving practical training times. The approach offers a scalable, theoretically grounded method for data-driven MILP modeling with potential applicability to scheduling, power systems, and related domains.

Abstract

In mixed-integer linear programming, data-driven inverse optimization that learns the objective function and the constraints from observed data plays an important role in constructing appropriate mathematical models for various fields, including power systems and scheduling. However, to the best of our knowledge, there is no known method for learning both the objective functions and the constraints. In this paper, we propose a two-stage method for a class of problems where the objective function is expressed as a linear combination of functions and the constraints are represented by functions and thresholds. Specifically, our method first learns the constraints and then learns the objective function. On the theoretical side, we show the proposed method can solve inverse optimization problems in finite dataset, develop statistical learning theory in pseudometric spaces and sub-Gaussian distributions, and construct a statistical learning for inverse optimization. On the experimental side, we demonstrate that our method is practically applicable for scheduling problems formulated as integer linear programmings with up to 100 decision variables, which are typical in real-world settings.

Inverse Mixed-Integer Programming: Learning Constraints then Objective Functions

TL;DR

This work addresses learning both the constraint set and the objective in MILP from observed decisions by proposing a two-stage inverse optimization framework that first learns constraint parameters and then objective weights. The forward problem is formulated as a MILP with a linear-combination objective and constraint maps defined by lattice-homomorphisms, enabling a tractable Solve_IOP learning procedure. The authors establish finite-data solvability, extend statistical learning theory to pseudometric spaces, and derive generalization bounds for inverse optimization, plus an empirical demonstration on a single-machine scheduling ILP with up to 100 decision variables achieving practical training times. The approach offers a scalable, theoretically grounded method for data-driven MILP modeling with potential applicability to scheduling, power systems, and related domains.

Abstract

In mixed-integer linear programming, data-driven inverse optimization that learns the objective function and the constraints from observed data plays an important role in constructing appropriate mathematical models for various fields, including power systems and scheduling. However, to the best of our knowledge, there is no known method for learning both the objective functions and the constraints. In this paper, we propose a two-stage method for a class of problems where the objective function is expressed as a linear combination of functions and the constraints are represented by functions and thresholds. Specifically, our method first learns the constraints and then learns the objective function. On the theoretical side, we show the proposed method can solve inverse optimization problems in finite dataset, develop statistical learning theory in pseudometric spaces and sub-Gaussian distributions, and construct a statistical learning for inverse optimization. On the experimental side, we demonstrate that our method is practically applicable for scheduling problems formulated as integer linear programmings with up to 100 decision variables, which are typical in real-world settings.

Paper Structure

This paper contains 48 sections, 57 theorems, 168 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Formulation of a class of inverse optimization Problems for MILP
  3. Learning constrains then objective functions
  4. Solvablity of inverse optimization problems for MILP
  5. Statistical learning theory in pseudometric spaces
  6. Statistical learning theory of inverse optimization
  7. Successful learning in integer linear programming with 100 decision bariables
  8. Related work
  9. Inverse optimization algorithms
  10. Loss functions for inverse optimization
  11. Learning constraints
  12. Learning objective functions
  13. Statistical learning thoery
  14. Background
  15. Lattice in Euclidean Space
  16. ...and 33 more sections

Key Result

Proposition 3.1

Let $\lambda > 0$ be a constant. For $x \in \mathcal{X}$, the following are equivalent: $x \in \mathrm{FOP}( \widetilde{\theta} , s )$ if and only if $\ell^{\mathrm{sub}, \lambda} ( x, \widetilde{\theta} , s ) = 0$.

Figures (2)

  • Figure : Minimization of suboptimality loss kitaoka2023proof
  • Figure : Maximizing feasible set then minimizing suboptimality loss

Theorems & Definitions (105)

  • Proposition 3.1: Cf. ren2025inverse
  • Proposition 3.3: kitaoka2024fast
  • Theorem 4.2
  • Remark 4.3
  • Example 4.4
  • Example 4.5
  • Proposition 4.6
  • Example 4.7
  • Remark 4.8
  • Remark 4.9
  • ...and 95 more