arxiv: 2605.06516 · v1 · submitted 2026-05-07 · 🧮 math.OC · cs.AI

Recognition: unknown

Learning to Cut: Reinforcement Learning for Benders Decomposition

Haochen Cai, Xian Yu

Pith reviewed 2026-05-08 08:04 UTC · model grok-4.3

classification 🧮 math.OC cs.AI

keywords Benders decompositionreinforcement learningcut selectionstochastic programmingREINFORCEpolicy gradientelectric vehicletwo-stage stochastic programs

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The pith

Reinforcement learning selects effective cuts to accelerate Benders decomposition for stochastic programs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Benders decomposition often slows as the master problem accumulates cuts when solving two-stage stochastic programs. The paper introduces RLBD, where a neural network policy learns to choose which cuts to generate and add at each step, trained end-to-end with the REINFORCE policy gradient algorithm. Tests on an electric vehicle charging station location problem show this approach reduces computation time compared to both standard Benders and a supervised learning baseline. The policy generalizes to problems with new data values and different numbers of decision variables. This points to a way of automating and improving a classic optimization technique for uncertain decision problems.

Core claim

RLBD adapts cut selection in Benders decomposition using a stochastic policy parameterized by a neural network and trained via REINFORCE. Numerical experiments on the two-stage stochastic electric vehicle charging station location problem demonstrate substantial improvements in computational efficiency over vanilla BD and LearnBD. The method also shows strong generalization capabilities to instances with similar structure but different data inputs and decision variable dimensions.

What carries the argument

A neural network that outputs a stochastic policy for selecting cuts, trained by the REINFORCE algorithm to minimize expected solution time in Benders decomposition.

Load-bearing premise

The benefits from faster solving during inference outweigh the upfront cost of training the policy network, and the policy generalizes beyond the electric vehicle test instances used for training.

What would settle it

If the RLBD method requires more total time including training or fails to reduce iterations on a held-out set of problems with different uncertainty distributions, the efficiency claim would not hold.

Figures

Figures reproduced from arXiv: 2605.06516 by Haochen Cai, Xian Yu.

**Figure 1.** Figure 1: The RLBD Framework. 4.1 Markov Decision Process Formulation We model the cut-selection problem in BD as a finite-horizon discounted MDP. In this formulation, each Benders iteration corresponds to one decision step. At iteration t, the agent observes the current master problem (RLBD-MPt): (RLBD-MPt) min x∈X,θ c Tx + X ω∈Ω p ω θ ω s.t. θ ω ≥ (π ω τ ) T (h ω − T ωx), ∀τ ∈ [t − 1], ω ∈ Aτ . (6) Unlike (MPt) in… view at source ↗

**Figure 2.** Figure 2: The shared neural network is applied to each candidate cut using the concatenation of view at source ↗

**Figure 3.** Figure 3: Hyperparameter tuning results under different combinations of view at source ↗

**Figure 4.** Figure 4: Evaluation results under different numbers of cuts added per iteration view at source ↗

**Figure 5.** Figure 5: Scenario-level exposure metrics sorted by scenario rank. view at source ↗

**Figure 6.** Figure 6: Evaluation results under different numbers of cuts added per iteration view at source ↗

read the original abstract

Benders decomposition (BD) is a widely used solution approach for solving two-stage stochastic programs arising in real-world decision-making under uncertainty. However, it often suffers from slow convergence as the master problem grows with an increasing number of cuts. In this paper, we propose Reinforcement Learning for BD (RLBD), a framework that adaptively selects cuts using a neural network-based stochastic policy. The policy is trained using a policy gradient method via the REINFORCE algorithm. We evaluate the proposed approach on a two-stage stochastic electric vehicle charging station location problem and compare it with vanilla BD and LearnBD, a supervised learning approach that classifies cuts using a support vector machine. Numerical results demonstrate that RLBD achieves substantial improvements in computational efficiency and exhibits strong generalization to problems with similar structures but varying data inputs and decision variable dimensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

RLBD applies REINFORCE to train a stochastic policy for cut selection in Benders decomposition and reports gains on an EV charging instance, but the generalization claim across decision-variable dimensions rests on unshown architecture details.

read the letter

The paper's core move is to frame cut addition in Benders decomposition as a sequential decision process and train a neural policy with the REINFORCE algorithm rather than using a supervised classifier like the earlier LearnBD work. That shift is the actual novelty. On the two-stage stochastic electric-vehicle charging location problem the method is said to produce faster convergence than both vanilla Benders and the SVM baseline while also generalizing to instances with different data and different numbers of variables. Those are the concrete claims worth checking. The setup itself is straightforward and the choice of policy gradient makes sense once you accept that cut quality is context-dependent and noisy. The application domain is also sensible; Benders is still the workhorse for many stochastic programs in energy and logistics, so any reliable speed-up would matter. The soft spot is the missing support for the generalization result. The abstract asserts strong transfer to problems with varying decision-variable dimensions, yet a standard feed-forward policy network takes a fixed-size state vector. Without an explicit dimension-agnostic representation (graph network, permutation-invariant pooling, or fixed feature map), any reported transfer is likely an artifact of testing on instances whose sizes are close to the training distribution. The abstract supplies no numbers, no statistical tests, no description of the state features, and no training-cost versus solve-time comparison, so the practical payoff cannot be judged yet. The training hyper-parameters are listed as free, which is honest but leaves the reproducibility burden on the reader. This work is aimed at people already running Benders on stochastic programs who are willing to add a learned component. A reader who wants to see RL applied to decomposition methods will find the framing useful even if the experiments need tightening. It is coherent on its own terms and engages the relevant prior literature, so it deserves a serious referee who can ask for the missing architectural and numerical details. I would send it out rather than desk-reject.

Referee Report

2 major / 0 minor

Summary. The paper introduces RLBD, a reinforcement learning framework that trains a neural network policy via REINFORCE to adaptively select cuts during Benders decomposition for two-stage stochastic programs. It is evaluated on a two-stage stochastic electric vehicle charging station location problem, where it is compared to vanilla BD and LearnBD (an SVM-based supervised approach), with claims of substantial computational efficiency gains and strong generalization to instances with similar structure but varying data inputs and decision variable dimensions.

Significance. If the empirical claims hold with proper controls, the work offers a novel RL-based mechanism for improving cut selection in Benders decomposition, which could reduce iteration counts and solve times for large-scale stochastic optimization problems common in energy and logistics applications. The policy-gradient training approach is a reasonable extension of existing learning-for-optimization ideas, though its practical impact depends on whether training overhead is amortized and whether the policy transfers beyond the single test family.

major comments (2)

[Abstract] Abstract: The central claim of 'strong generalization to problems with similar structures but varying data inputs and decision variable dimensions' is load-bearing for the paper's practical contribution, yet the described experiments are confined to the electric-vehicle charging instance family. No quantitative results, problem sizes, or architecture details (e.g., input feature construction or use of GNN/permutation-invariant layers) are supplied to show that the policy network can be applied to instances with different numbers of decision variables without retraining or ad-hoc padding.
[Abstract] Abstract and evaluation description: No quantitative metrics, statistical tests, problem dimensions, training hyperparameters, or wall-clock comparisons (including policy training cost) are provided to support the asserted 'substantial improvements in computational efficiency.' This absence prevents verification of whether the RL policy actually outperforms the baselines on the reported instances or merely fits the training distribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments. We address the major points below and will revise the manuscript accordingly to strengthen the presentation of our results and claims.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim of 'strong generalization to problems with similar structures but varying data inputs and decision variable dimensions' is load-bearing for the paper's practical contribution, yet the described experiments are confined to the electric-vehicle charging instance family. No quantitative results, problem sizes, or architecture details (e.g., input feature construction or use of GNN/permutation-invariant layers) are supplied to show that the policy network can be applied to instances with different numbers of decision variables without retraining or ad-hoc padding.

Authors: We agree that the abstract would benefit from additional detail on these aspects. The full manuscript evaluates the policy on the EV charging family across multiple instance sizes, including varying numbers of charging stations (decision variables) and scenarios, without retraining the policy. The network uses a fixed set of aggregated structural features (e.g., statistics over candidate cuts and scenario costs) that are independent of exact dimension, enabling direct transfer. We will revise the abstract to report specific problem dimensions tested, quantitative generalization metrics (e.g., performance on held-out sizes), and a concise description of the feature construction and network architecture. revision: yes
Referee: [Abstract] Abstract and evaluation description: No quantitative metrics, statistical tests, problem dimensions, training hyperparameters, or wall-clock comparisons (including policy training cost) are provided to support the asserted 'substantial improvements in computational efficiency.' This absence prevents verification of whether the RL policy actually outperforms the baselines on the reported instances or merely fits the training distribution.

Authors: The full paper reports iteration counts, solve times, and direct comparisons to vanilla BD and LearnBD on the EV instances, with the RL policy showing lower iteration counts and faster overall solution times. We acknowledge that the abstract omits these specifics along with hyperparameters and training overhead. We will revise the abstract to include key quantitative results (e.g., average reductions in iterations and wall-clock time), problem dimensions, a note on multiple random seeds for statistical reliability, and clarification that reported times include amortized training cost, which is offset by repeated use on similar instances. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical RL training and held-out evaluation are independent of claimed results

full rationale

The paper trains a stochastic policy via REINFORCE on electric-vehicle instances and reports speedups plus generalization on separate test instances with varying inputs and dimensions. No derivation step reduces a 'prediction' to a fitted parameter by construction, no self-definitional equations appear, and no load-bearing claim rests on self-citation. The numerical improvements are direct experimental outcomes on held-out data, satisfying the self-contained benchmark criterion.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on standard RL convergence assumptions and the structure of Benders cuts; no new entities are postulated.

free parameters (1)

neural network architecture and training hyperparameters
The policy network weights and learning settings are fitted during REINFORCE training on the EV instances; exact values and architecture not stated in abstract.

axioms (1)

standard math REINFORCE policy gradient converges under standard conditions on the MDP formulation of cut selection
Invoked implicitly when claiming the trained policy improves efficiency.

pith-pipeline@v0.9.0 · 5426 in / 1134 out tokens · 76305 ms · 2026-05-08T08:04:27.452945+00:00 · methodology

discussion (0)

Reference graph

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