# WeCon: An Efficient Weight-Conditioned Neural Solver for Multi-Objective Combinatorial Optimization Problems

arXiv: https://arxiv.org/abs/2605.22876 · HTML: https://mlsanity.com/paper/2605.22876/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 54 references that are in the map.
- First posted on arXiv 20 May 2026.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [Deep Reinforcement Learning for Multi-objective Optimization](https://mlsanity.com/paper/1906.02386/) (2019)
- [MVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts](https://mlsanity.com/paper/2405.01029/) (2024)
- [Towards Generalizable Neural Solvers for Vehicle Routing Problems via Ensemble with Transferrable Local Policy](https://mlsanity.com/paper/2308.14104/) (2023)
- [Pareto Set Learning for Neural Multi-objective Combinatorial Optimization](https://mlsanity.com/paper/2203.15386/) (2022)
- [RouteFinder: Towards Foundation Models for Vehicle Routing Problems](https://mlsanity.com/paper/2406.15007/) (2024)
- [CaDA: Cross-Problem Routing Solver with Constraint-Aware Dual-Attention](https://mlsanity.com/paper/2412.00346/) (2024)
- [Neural Combinatorial Optimization Algorithms for Solving Vehicle Routing Problems: A Comprehensive Survey with Perspectives](https://mlsanity.com/paper/2406.00415/) (2024)
- [Efficient Meta Neural Heuristic for Multi-Objective Combinatorial Optimization](https://mlsanity.com/paper/2310.15196/) (2023)
- [Meta-Learning-Based Deep Reinforcement Learning for Multiobjective Optimization Problems](https://mlsanity.com/paper/2105.02741/) (2021)
- [Distilling Autoregressive Models to Obtain High-Performance Non-Autoregressive Solvers for Vehicle Routing Problems with Faster Inference Speed](https://mlsanity.com/paper/2312.12469/) (2023)
- [Equity-Transformer: Solving NP-hard Min-Max Routing Problems as Sequential Generation with Equity Context](https://mlsanity.com/paper/2306.02689/) (2023)
- [BOPO: Neural Combinatorial Optimization via Best-anchored and Objective-guided Preference Optimization](https://mlsanity.com/paper/2503.07580/) (2025)
- [MODRL/D-AM: Multiobjective Deep Reinforcement Learning Algorithm Using Decomposition and Attention Model for Multiobjective Optimization](https://mlsanity.com/paper/2002.05484/) (2020)
- [Preference Optimization for Combinatorial Optimization Problems](https://mlsanity.com/paper/2505.08735/) (2025)
- [An Efficient Diffusion-based Non-Autoregressive Solver for Traveling Salesman Problem](https://mlsanity.com/paper/2501.13767/) (2025)
- [SHIELD: Multi-task Multi-distribution Vehicle Routing Solver with Sparsity and Hierarchy](https://mlsanity.com/paper/2506.08424/) (2025)
- [Preference-Driven Multi-Objective Combinatorial Optimization with Conditional Computation](https://mlsanity.com/paper/2506.08898/) (2025)
- [GELD: A Unified Neural Model for Efficiently Solving Traveling Salesman Problems Across Different Scales](https://mlsanity.com/paper/2506.06634/) (2025)
- [Efficient Neural Combinatorial Optimization Solver for the Min-max Heterogeneous Capacitated Vehicle Routing Problem](https://mlsanity.com/paper/2507.21386/) (2025)
- [Learning to Solve Combinatorial Optimization under Positive Linear Constraints via Non-Autoregressive Neural Networks](https://mlsanity.com/paper/2409.04495/) (2024)
- [Diversity Optimization for Travelling Salesman Problem via Deep Reinforcement Learning](https://mlsanity.com/paper/2501.00884/) (2025)
- [VAGPO: Vision-augmented Asymmetric Group Preference Optimization for Graph Routing Problems](https://mlsanity.com/paper/2508.01774/) (2025)

## Abstract (arXiv)

Existing neural solvers for Multi-Objective Combinatorial Optimization Problems (MOCOPs) commonly adopt decomposition-based strategies that scalarize a MOCOP into multiple subproblems associated with distinct weight vectors. However, they either inject weights only once during decoding, limiting weight-conditioned context modeling, or primarily during encoding, causing weight-signal dilution during decoding. Moreover, their preference optimization methods rely on purely random sampling to construct solution pairs for training solvers, which often produces less informative pairs and thus leads to low training effectiveness. To better address these limitations, we propose an efficient Weight-Conditioned neural solver (WeCon). Specifically, we design an encoder layer with three attention blocks and our proposed Gated Residual Fusion block to facilitate harmonious interaction between instance features and weights, thereby generating informative weight-conditioned context. We further introduce a plug-and-play Residual Fusion block in the decoder to alleviate weight-signal dilution. Finally, we propose Efficient Preference Optimization, which constructs high-quality solutions, thereby generating more informative pairs to improve training effectiveness. Experimental results on four MOCOP variants across different problem scales and distribution patterns demonstrate that WeCon achieves HyperVolume (HV) performance comparable to the state-of-the-art (SOTA) solver POCCO-W, while requiring approximately 40% less inference time. Moreover, the variant WeCon-CCO, which adopts an enhanced decoder, achieves the best overall HV performance with increased inference time. Ablation studies validate the contributions of all proposed designs.