The Knapsack Problem is a classic problem in combinatorial optimisation. Solving these problems may be computationally expensive. Recent years have seen a growing interest in the use of deep learning methods to approximate the solutions to such problems. A core problem is how to enforce or encourage constraint satisfaction in predicted solutions. A promising approach for predicting solutions to constrained optimisation problems is the Lagrangian Dual Framework which builds on the method of Lagrangian Relaxation. In this paper we develop neural network models to approximate Knapsack Problem solutions using the Lagrangian Dual Framework while improving constraint satisfaction. We explore the problems of output interpretation and model selection within this context. Experimental results show strong constraint satisfaction with a minor reduction of optimality as compared to a baseline neural network which does not explicitly model the constraints.

翻译：背包问题是组合优化领域的经典问题。求解此类问题可能计算成本高昂。近年来，利用深度学习方法近似求解此类问题日益受到关注。其核心难题在于如何在预测解中强制或鼓励约束条件的满足。拉格朗日对偶框架基于拉格朗日松弛方法，为预测约束优化问题的解提供了一种前景广阔的方法。本文利用拉格朗日对偶框架开发神经网络模型，在提升约束满足性的同时近似求解背包问题。我们探讨了在该框架下的输出解释与模型选择问题。实验结果表明，与未显式建模约束的基线神经网络相比，本方法在保持较强约束满足性的同时，仅带来轻微的最优性损失。