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Lagrangian cuts generated by batch to efficiently solve two-stage stochastic mixed-integer program

Luo Xiaoyu, Gao Chuanhou

TL;DR

Computational study demonstrates that the proposed algorithm can significantly improve the lower bound of the linear relaxation of the Bender master problem more quickly with much fewer Lagrangian cuts.

Abstract

We propose to generate Lagrangian cut for two-stage stochastic integer program by batch, in contrast to the existing methods which solve each Lagrangian subproblem at every iteration. We establish two convergence properties of the proposed algorithm. Then we demonstrate that the improvement in the lower bound achieved by incorporating the Lagrangian cut adheres to the `triangle inequality', thereby showcasing the superiority of our proposed method over existing approaches. Moreover, we suggest acquiring Lagrangian cuts for unresolved scenarios by averaging the coefficients of the acquired Lagrangian cuts, ensuring the quality of this cut with a certain probability. Computational study demonstrates that our proposed algorithm can significantly improve the lower bound of the linear relaxation of the Bender master problem more quickly with much fewer Lagrangian cuts.

Lagrangian cuts generated by batch to efficiently solve two-stage stochastic mixed-integer program

TL;DR

Computational study demonstrates that the proposed algorithm can significantly improve the lower bound of the linear relaxation of the Bender master problem more quickly with much fewer Lagrangian cuts.

Abstract

We propose to generate Lagrangian cut for two-stage stochastic integer program by batch, in contrast to the existing methods which solve each Lagrangian subproblem at every iteration. We establish two convergence properties of the proposed algorithm. Then we demonstrate that the improvement in the lower bound achieved by incorporating the Lagrangian cut adheres to the `triangle inequality', thereby showcasing the superiority of our proposed method over existing approaches. Moreover, we suggest acquiring Lagrangian cuts for unresolved scenarios by averaging the coefficients of the acquired Lagrangian cuts, ensuring the quality of this cut with a certain probability. Computational study demonstrates that our proposed algorithm can significantly improve the lower bound of the linear relaxation of the Bender master problem more quickly with much fewer Lagrangian cuts.
Paper Structure (29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem \siamprelabel1

Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure \siamprelabel1: Example figure using external image files.
  • Figure \siamprelabel2: Example PGFPLOTS figure.

Theorems & Definitions (4)

  • Theorem \siamprelabel1: Mean Value Theorem
  • Corollary \siamprelabel2
  • Proof 1
  • Proof 2: Proof of main theorem