A Stochastic Benders Decomposition Scheme for Large-Scale Stochastic Network Design

TL;DR

This work proposes a stochastic variant of Benders decomposition that mitigates the high computational cost of generating each cut by sampling a subset of the data at each iteration and nonetheless, generates deterministically valid cuts, via a dual averaging technique, rather than the probabilistically valid cuts frequently proposed in the stochastic optimization literature.

Abstract

Network design problems involve constructing edges in a transportation or supply chain network to minimize construction and daily operational costs. We study a stochastic version where operational costs are uncertain due to fluctuating demand and estimated as a sample average from historical data. This problem is computationally challenging, and instances with as few as 100 nodes often cannot be solved to optimality using current decomposition techniques. We propose a stochastic variant of Benders decomposition that mitigates the high computational cost of generating each cut by sampling a subset of the data at each iteration and nonetheless generates deterministically valid cuts, rather than the probabilistically valid cuts frequently proposed in the stochastic optimization literature, via a dual averaging technique. We implement both single-cut and multi-cut variants of this Benders decomposition, as well as a variant that uses clustering of the historical scenarios. To our knowledge, this is the first single-tree implementation of Benders decomposition that facilitates sampling. On instances with 100-200 nodes and relatively complete recourse, our algorithm achieves 5-7% optimality gaps, compared with 16-27% for deterministic Benders schemes, and scales to instances with 700 nodes and 50 commodities within hours. Beyond network design, our strategy could be adapted to generic two-stage stochastic mixed-integer optimization problems where second-stage costs are estimated via a sample average.

Figures (8)

• Figure 1: Distribution of the number of outer-loop iterations required by the single-stochastic cutting-plane algorithms with single-cut root node cuts on small-scale (left panel) and medium-scale (right panel) instances; see Table \ref{['tab:problem_sizes']} for definitions of small and medium-scale instances.
• Figure 1: Effect of Regularizer on the Algorithm's Performance.
• Figure 1: Optimality gaps achieved by the single-cut stochastic cutting plane algorithm on all synthetic instances. For each combination of number of nodes $|\mathcal{N}|$, number of commodities $|\mathcal{K}|$, and number of scenarios $|\mathcal{R}|$, results are averaged across 3 random instances.
• Figure 2: Impact of implementing a cutting-plane algorithm with one second-stage variable $\boldsymbol{x}$ in the master problem (to ensure at least feasibility for the average demand) on the average optimality gap achieved on the R instances, as the number of scenarios $|\mathcal{R}|$ increases. Bars represent standard errors.
• Figure 2: Optimality gaps achieved by the accelerated multi-cut stochastic cutting plane algorithm on all synthetic instances. For each combination of number of nodes $|\mathcal{N}|$, number of commodities $|\mathcal{K}|$, and number of scenarios $|\mathcal{R}|$, results are averaged across 3 random instances.

Theorems & Definitions (7)

• Proposition 1
• Remark 1
• Proposition 2
• Remark 2
• Remark 3
• Proposition A.1
• Proposition A.2