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Learning Valid Dual Bounds in Constraint Programming: Boosted Lagrangian Decomposition with Self-Supervised Learning

Swann Bessa, Darius Dabert, Max Bourgeat, Louis-Martin Rousseau, Quentin Cappart

TL;DR

A self-supervised learning approach that leverages neural networks to generate multipliers directly, yielding tight bounds in constraint programming is proposed, which significantly reduces the number of sub-gradient optimization steps required, enhancing the pruning efficiency and reducing the execution time of constraint programming solvers.

Abstract

Lagrangian decomposition (LD) is a relaxation method that provides a dual bound for constrained optimization problems by decomposing them into more manageable sub-problems. This bound can be used in branch-and-bound algorithms to prune the search space effectively. In brief, a vector of Lagrangian multipliers is associated with each sub-problem, and an iterative procedure (e.g., a sub-gradient optimization) adjusts these multipliers to find the tightest bound. Initially applied to integer programming, Lagrangian decomposition also had success in constraint programming due to its versatility and the fact that global constraints provide natural sub-problems. However, the non-linear and combinatorial nature of sub-problems in constraint programming makes it computationally intensive to optimize the Lagrangian multipliers with sub-gradient methods at each node of the tree search. This currently limits the practicality of LD as a general bounding mechanism for constraint programming. To address this challenge, we propose a self-supervised learning approach that leverages neural networks to generate multipliers directly, yielding tight bounds. This approach significantly reduces the number of sub-gradient optimization steps required, enhancing the pruning efficiency and reducing the execution time of constraint programming solvers. This contribution is one of the few that leverage learning to enhance bounding mechanisms on the dual side, a critical element in the design of combinatorial solvers. To our knowledge, this work presents the first generic method for learning valid dual bounds in constraint programming.

Learning Valid Dual Bounds in Constraint Programming: Boosted Lagrangian Decomposition with Self-Supervised Learning

TL;DR

A self-supervised learning approach that leverages neural networks to generate multipliers directly, yielding tight bounds in constraint programming is proposed, which significantly reduces the number of sub-gradient optimization steps required, enhancing the pruning efficiency and reducing the execution time of constraint programming solvers.

Abstract

Lagrangian decomposition (LD) is a relaxation method that provides a dual bound for constrained optimization problems by decomposing them into more manageable sub-problems. This bound can be used in branch-and-bound algorithms to prune the search space effectively. In brief, a vector of Lagrangian multipliers is associated with each sub-problem, and an iterative procedure (e.g., a sub-gradient optimization) adjusts these multipliers to find the tightest bound. Initially applied to integer programming, Lagrangian decomposition also had success in constraint programming due to its versatility and the fact that global constraints provide natural sub-problems. However, the non-linear and combinatorial nature of sub-problems in constraint programming makes it computationally intensive to optimize the Lagrangian multipliers with sub-gradient methods at each node of the tree search. This currently limits the practicality of LD as a general bounding mechanism for constraint programming. To address this challenge, we propose a self-supervised learning approach that leverages neural networks to generate multipliers directly, yielding tight bounds. This approach significantly reduces the number of sub-gradient optimization steps required, enhancing the pruning efficiency and reducing the execution time of constraint programming solvers. This contribution is one of the few that leverage learning to enhance bounding mechanisms on the dual side, a critical element in the design of combinatorial solvers. To our knowledge, this work presents the first generic method for learning valid dual bounds in constraint programming.
Paper Structure (28 sections, 16 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 28 sections, 16 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Cost-Based Domain Filtering.
  4. CP-based Lagrangian Relaxation.
  5. Lagrangian Decomposition in CP.
  6. Learning Valid Dual Bounds.
  7. Statement of Contributions.
  8. Lagrangian Decomposition in CP
  9. Learning Lagrangian Multipliers
  10. Graph Neural Network Architecture
  11. Case studies
  12. Multi-dimensional Knapsack Problem (MKP).
  13. Shift Scheduling Problem (SSP).
  14. Experimental Protocol
  15. Competitors.
  16. ...and 13 more sections

Figures (3)

  • Figure 1: Main steps of LD in CP. Red elements illustrate the standard sub-gradient optimization, while the blue elements highlight our contribution based on self-supervised learning.
  • Figure 2: Relative performance of our approaches compared to the CP+SG baseline in execution time. Each dot below 0% indicates a reduction in execution time with our method for a specific instance.
  • Figure 3: Evolution of the bounds for the two most challenging configurations (gap with the optimal solution).