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Accelerating Benders decomposition for solving a sequence of sample average approximation replications

Harshit Kothari, James R. Luedtke

TL;DR

This paper tackles accelerating the solution of sequences of sample-average approximations for two-stage stochastic programs with continuous recourse by reusing information across replications. It introduces a dual solution pool (DSP), a curated DSP, and initialization techniques (static and adaptive) to speed up Benders decomposition, both for LPs and IPs, across CFLP, CMND, and UC problem classes. Empirical results show substantial time savings (roughly 5–10x overall, up to 10x in some cases) when using these information reuse strategies, with adaptive initialization delivering the strongest improvements, especially for IPs. The methods preserve solution quality while dramatically reducing subproblem solves and cuts, highlighting practical impact for obtaining confidence intervals and statistical estimates via multiple SAA replications in stochastic programming.

Abstract

Sample average approximation (SAA) is a technique for obtaining approximate solutions to stochastic programs that uses the average from a random sample to approximate the expected value that is being optimized. Since the outcome from solving an SAA is random, statistical estimates on the optimal value of the true problem can be obtained by solving multiple SAA replications with independent samples. We study techniques to accelerate the solution of this set of SAA replications, when solving them sequentially via Benders decomposition. We investigate how to exploit similarities in the problem structure, as the replications just differ in the realizations of the random samples. Our extensive computational experiments provide empirical evidence that our techniques for using information from solving previous replications can significantly reduce the solution time of later replications.

Accelerating Benders decomposition for solving a sequence of sample average approximation replications

TL;DR

This paper tackles accelerating the solution of sequences of sample-average approximations for two-stage stochastic programs with continuous recourse by reusing information across replications. It introduces a dual solution pool (DSP), a curated DSP, and initialization techniques (static and adaptive) to speed up Benders decomposition, both for LPs and IPs, across CFLP, CMND, and UC problem classes. Empirical results show substantial time savings (roughly 5–10x overall, up to 10x in some cases) when using these information reuse strategies, with adaptive initialization delivering the strongest improvements, especially for IPs. The methods preserve solution quality while dramatically reducing subproblem solves and cuts, highlighting practical impact for obtaining confidence intervals and statistical estimates via multiple SAA replications in stochastic programming.

Abstract

Sample average approximation (SAA) is a technique for obtaining approximate solutions to stochastic programs that uses the average from a random sample to approximate the expected value that is being optimized. Since the outcome from solving an SAA is random, statistical estimates on the optimal value of the true problem can be obtained by solving multiple SAA replications with independent samples. We study techniques to accelerate the solution of this set of SAA replications, when solving them sequentially via Benders decomposition. We investigate how to exploit similarities in the problem structure, as the replications just differ in the realizations of the random samples. Our extensive computational experiments provide empirical evidence that our techniques for using information from solving previous replications can significantly reduce the solution time of later replications.

Paper Structure

This paper contains 34 sections, 1 theorem, 33 equations, 5 figures, 23 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Benders Decomposition
  3. Stochastic LPs
  4. Stochastic IPs
  5. Techniques for Reusing Information to Accelerate Benders Decomposition
  6. Dual Solution Pool
  7. Curated DSP
  8. Initialization Techniques
  9. Static Initialization
  10. Adaptive Initialization
  11. Phase One.
  12. Phase two.
  13. Extensions
  14. Relatively Complete Recourse
  15. Single-Cut
  16. ...and 19 more sections

Key Result

lemma thmcounterlemma

Let $\mathbf{c}^i$ for $i = 1,\ldots,N$ and $\hat{\mathbf{c}}$ be random coefficient vectors drawn independently from an identical distribution $\mathcal{D}$ on $\mathbb{R}^d$. Let $\epsilon > 0$ and let $M(\Pi,\epsilon)$ be the covering number of $\Pi$ with respect to the Euclidean norm at scale $\

Figures (5)

  • Figure 1: Plots showing the fraction of solved LP instances over time for CFLP, CMND, and UC problems.
  • Figure 2: Plots showing the fraction of solved LP instances over time for CFLP, CMND, and UC problems.
  • Figure 3: Total time and root gap trends for boosted static initialization and adaptive initialization per replication. The CFLP results (top plots) are for the instance with 35 facilities and 105 customers. The CMND results (bottom plots) are for the r03.3 instance.
  • Figure 4: Number of dual solutions in the pool for DSP and curated DSP.
  • Figure 5: Comparison of performance of methods as number of scenarios vary.

Theorems & Definitions (2)

  • lemma thmcounterlemma
  • proof