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LOTUS: A Warm-Start Framework for Powering Dual Decomposition in Large-Scale Two-Stage Stochastic Programs

Emma Cornielje, Berend Markhorst, Alessandro Zocca, Rob van der Mei

TL;DR

LOTS is proposed, a subset-based warm-start framework that enhances Dual Decomposition under fixed time budgets and accelerates primal convergence and partially alleviates the impact of weak LP relaxations by initializing the dual search with informed multipliers.

Abstract

Solving large two-stage stochastic mixed-integer programs is computationally challenging. We propose LOTUS, a subset-based warm-start framework that enhances Dual Decomposition under fixed time budgets. By initializing the dual search with informed multipliers, LOTUS accelerates primal convergence and partially alleviates the impact of weak LP relaxations. Through an extensive computational study on production planning instances, we show that, within two hours, LOTUS yields significantly better primal solutions in 45.83% of cases, while being outperformed by Dual Decomposition in only 4.17%.

LOTUS: A Warm-Start Framework for Powering Dual Decomposition in Large-Scale Two-Stage Stochastic Programs

TL;DR

LOTS is proposed, a subset-based warm-start framework that enhances Dual Decomposition under fixed time budgets and accelerates primal convergence and partially alleviates the impact of weak LP relaxations by initializing the dual search with informed multipliers.

Abstract

Solving large two-stage stochastic mixed-integer programs is computationally challenging. We propose LOTUS, a subset-based warm-start framework that enhances Dual Decomposition under fixed time budgets. By initializing the dual search with informed multipliers, LOTUS accelerates primal convergence and partially alleviates the impact of weak LP relaxations. Through an extensive computational study on production planning instances, we show that, within two hours, LOTUS yields significantly better primal solutions in 45.83% of cases, while being outperformed by Dual Decomposition in only 4.17%.
Paper Structure (44 sections, 14 equations, 2 figures, 4 tables)

This paper contains 44 sections, 14 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Literature
  3. Decomposition Frameworks
  4. Accelerating Convergence: Warm-Starts and Reduction
  5. Methodology
  6. Method
  7. Step 1: Dual Decomposition
  8. Step 2: Scenario Reduction (Warm Start)
  9. Distribution-based metric
  10. Problem-based metric
  11. Combining the metrics
  12. Selection algorithm
  13. Step 3: Solve the Reduced Lagrangian Dual
  14. Step 4: Warm Start Initialization
  15. Step 5: Solve the Full Lagrangian Dual
  16. ...and 29 more sections

Figures (2)

  • Figure 1: Schematic representation of LOTUS. The dashed arrows correspond to steps in this method. The red, orange, and green boxes denote the original, the full relaxed and the reduced relaxed problem, respectively.
  • Figure 2: Aggregated primal bound ratio ($R = Z_{P,LOTUS} / Z_{P,DD}$) over time (in blue) and percentage of instances in the warm-start phase over time (in red).