Title: Adaptive Constrained Optimization for Neural Vehicle Routing
Venue: NeurIPS 2025
Accepted: no
Keywords: Neural combinatorial optimization

Reinforcement learning

Constrained optimization
Abstract: Neural solvers have shown remarkable success in tackling Vehicle Routing Problems (VRPs). However, their application to scenarios with complex real-world constraints is still at an early stage. Recent works successfully employ variants of the Lagrange multiplier method to handle such constraints, but their limitation lies in the use of a uniform multiplier across all problem instances, overlooking the fact that the difficulty of satisfying constraints varies significantly across instances. To address this limitation, we propose an instance-level adaptive constrained optimization framework that reformulates the Lagrangian dual problem by assigning each instance its own multiplier. To efficiently optimize this new problem, we design a multiplier-conditioned policy that solves instances with a controllable level of constraint awareness, which effectively decouples policy optimization from the optimization of multipliers. By leveraging this conditioned policy, we customize the optimization of multipliers for each test instance by adapting to its particular constraint violations. Experimental results on the Travelling Salesman Problem with Time Window (TSPTW), and TSP with Draft Limit (TSPDL) show that our method exhibits advantages compared to the strong solver LKH3 and significantly outperforms state-of-the-art neural methods.


Peer reviews:
comments: SUMMARY:
This paper proposes an two-stage adaptive constraint optimization method, which reexpresses the Lagrange dual problem of combinatorial optimization problems by assigning its own multipliers to each specific instance. The algorithm has achieved better results on two special cases with more constraints in the TSP problem.

STRENGTHS AND WEAKNESSES:
Strength:
1.This paper raises a profound issue. The coefficient design of the Lagrange multiplier method depends on each problem, and this paper proposes a method to solve it, which is of great significance.
2.The author completed sufficient ablation experiments, proving that multiple designs in the proposed model are important.
3.The proposed network architecture is easy to read.
Weakness:
1.It seems that the article does not provide the specific calculation method of HV, which has raised some doubts in me about the validity of HV. Except for the TSPDL50 problem where the feasible Rate and Average Gap cannot achieve simultaneous SOTA, if there is no convincing HV formula, the persuasiveness of this comparison is insufficient.
2.The scale of the questions for comparison is limited. Increasing the scale of the problem, adding the types of sampling, and adopting real datasets such as TSBLIB may better solve this problem.

QUESTIONS:
See Weaknesses. Mainly Weakness 1. If you inform me of the specific formula of HV, or if you give the performance of the source data based on multiple distributions or real datasets like TSPLIB, I may improve my score.
Small issue:
1.What is the full name of PIP? Please give its full name when first appear in the text, no matter whether it is the content of the important literature you have cited or not.

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-12

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper builds upon the PIP model introduced in \"Learning to Handle Complex Constraints for Vehicle Routing Problems\", with a focus on addressing routing challenges such as TSPTW and TSPDL, where masks alone cannot easily prevent infeasible solutions. The key advancement lies in iteratively adjusting Lagrangian multipliers at the instance level within the PIP framework. This approach enhances the optimization of the dual objectives of solution feasibility and objective function value. By employing subgradient methods, the paper introduces an innovative solution to improve upon the fixed Lagrangian multipliers in the original PIP model.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n1. The paper is well-organized, following a logical progression from motivation to analysis, methodology, and experimental validation. This structure significantly enhances readability and comprehension.\n2. The proposed method demonstrates notable improvements in learning-based approaches for problems like TSPTW and TSPDL, where constraints cannot be directly managed through masks. This advancement broadens the applicability of learning-based methods in complex routing scenarios.\n\n## Weaknesses\n1. The paper's innovation primarily revolves around the dynamic iterative improvement and adjustment of fixed Lagrangian multipliers in the PIP model. This singular focus may limit the perceived novelty of the contribution.\n2. While the method shows promise, it does not consistently outperform the PIP model. Specifically, on the TSPTW50 dataset, the objective function value offers no advantage, and on the TSPDL100 dataset, instance feasibility shows no improvement. Furthermore, the lack of comparison with the PIP model on CVRP datasets, despite acknowledged implementation challenges, weakens the overall persuasiveness of the results. The paper also fails to demonstrate significant superiority on other benchmark datasets.\n3. The paper does not adequately explore the model's generalization capabilities, leaving questions about its applicability to a wider range of problems or variations.\n\nQUESTIONS:\n1. **Comparison with PIP-D**: The original paper \"Learning to Handle Complex Constraints for Vehicle Routing Problems\" also proposed a PIP-D model, which achieved state-of-the-art results in most cases. Why does this paper not include comparisons with PIP-D? This omission may leave a gap in understanding the full scope of the model's performance.\n2. **Training Efficiency**: During training, the model introduces an additional parameter lambda to enhance the base model's generalization ability regarding lambda. To strengthen the credibility of the proposed model, it would be valuable to present:\n   - The training time required for a single lambda to achieve comparable effectiveness to the PIP model.\n   - The training time needed for multiple different lambdas of the base model to match the performance of the PIP model.\n   - Comparative experiments between the base model and the enhanced model to provide a comprehensive assessment of training efficiency and effectiveness.\n3. **Generalization During Inference**: During inference or fine-tuning, as lambda iterates, if a lambda value not encountered during training appears, does the model possess sufficient capacity to adapt and learn effectively?\n\nLIMITATIONS:\nMore executing time may be required during the inference.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-19"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces ICO, an optimization framework designed to address the limitations of traditional NCO methods, where the Lagrange multipliers are fixed. Experimental results demonstrate that ICO yields promising results, particularly for the TSPTW and TSPDL.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n1. The paper enhances the constraint handling capability of neural VRP solvers, which is an important contribution to the field.\n2. The motivation for the proposed method is well-articulated, and it is coherently integrated with the methodology, providing a clear rationale for the approach.\n3. The paper is well-structured and easy to follow, with a logical flow between sections.\n\n**Weakness:**\n1. The method is primarily based on the specific model PIP, and does not seem to demonstrate generality for different models (e.g., BQ and GOAL). Furthermore, the performance gain (compared to PIP) achieved is not significant.\n\n2. As the experiments (include three difficulty levels) conducted in PIP paper, the evaluation of this work is only conducted under the \"hard\" difficulty level, which limits the generalizability of the findings.\n\n3. The experimental section lacks clarity in certain areas. For example, the default setting for λ in POMO+PIP and how it was configured for the TSPTW / DL of size 50 are not specified.\n\n4. The proposed approach is heuristic in nature, and there is no theoretical guarantee or proof of its effectiveness, which raises concerns about its long-term reliability and robustness.\n\nQUESTIONS:\n1. Can the authors provide experiments of ICO under the other two difficulty levels, as PIP does?\n\n2. The paper uses the \"instance infeasibility rate\" but does not consider \"solution infeasibility rate.\" Could the authors explain why the latter was excluded?\n\n3. Can ICO be applied to other algorithms based on Lagrange multipliers? A discussion on this would clarify its broader applicability.\n\n4. How does the training time of ICO compare to PIP?\n\nLIMITATIONS:\nN/A.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the problem of solving Vehicle Routing Problems (VRPs) with complex constraints, such as time windows, using reinforcement learning (RL). A common approach for solving such constrained optimization problems is the Lagrange multiplier method, which can be combined with RL by introducing a constraint-violation term into the rewards with a multiplier $\\lambda$. This paper focuses on the problem of determining the value of the multiplier $\\lambda$ for each instance during training, and learning policies that can be conditioned on instance-specific $\\lambda$ values. The proposed approach includes pre-training with random $\\lambda$ and fine-tuning with adaptive $\\lambda$ values, and it uses an extension of POMO [41] as the RL policy. The paper presents numerical results showing improvement over existing approaches in terms of solution quality and constraint satisfaction.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The paper considers an interesting problem with practical applications.\n* The proposed approach is described clearly. Generally, the paper is well written and the formalisms are sound.\n* The paper provides sufficient details for reproducibility, and the source code for the experiments is available.\n* The numerical results are promising, showing improvement over prior work.\n* The numerical results also include ablation studies.\n\nWeaknesses:\n* The adaptive update of the multiplier $\\lambda$ in the fine-tuning stage can only increase $\\lambda$ but never decrease it. This seems suboptimal since policy updates may necessitate reduction to $\\lambda$. There are experimental results in the appendix on a more advanced, PID-based update of $\\lambda$, but these experiments apply the PID-based update only during inference. It is not clear why this approach was not tried during training since it seems like a very intuitive and obvious idea.\n* The proposed approach still requires adaptive tuning of $\\lambda$ for each problem instance, both during training and inference. The discussion of future work mentions the idea of training a predictor that can quickly predict a (quasi-)optimal $\\lambda$ value for a problem instance. This again seems like an obvious idea that should be tried. Including such a predictor (and perhaps training it using an RL paradigm) would make the paper much stronger.\n* It is not clear what the \"number of timeout nodes\" is and why it is included in the penalty term.\n* The paper could provide more guidance on how the random distribution of $\\lambda$ values should be selected for the pre-training stage.\n* \"...these two problems pose greater challenges in satisfying constraints compared to CVRPTW and CVRPDL, as the constraint violations of the latter can be addressed more easily by assigning additional vehicles to the violated nodes\"\nCVRPTW and CVRPDL may have limited number of vehicles, in which case the constraint violations cannot be addressed so easily.\n* The copyright file of the source code seems to de-anonymize the paper.\n\nQUESTIONS:\nIs there a reason for using REINFORCE instead of more advanced algorithms, such as PPO?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}