In many applications, a combinatorial problem must be repeatedly solved with similar, but distinct parameters. Yet, the parameters $w$ are not directly observed; only contextual data $d$ that correlates with $w$ is available. It is tempting to use a neural network to predict $w$ given $d$. However, training such a model requires reconciling the discrete nature of combinatorial optimization with the gradient-based frameworks used to train neural networks. When the problem in question is an Integer Linear Program (ILP), one approach to overcome this training issue is to consider a continuous relaxation of the combinatorial problem. While existing methods utilizing this approach have shown to be highly effective on small problems, they do not always scale well to large problems. In this work, we draw on ideas from modern convex optimization to design a network and training scheme which scales effortlessly to problems with thousands of variables. Our experiments verify the computational advantage our proposed method enjoys on two representative problems, namely the shortest path problem and the knapsack problem.

翻译：在许多应用中，需要反复求解具有相似但不同参数的组合优化问题。然而，参数$w$并非直接可观测，仅能获得与$w$相关的上下文数据$d$。利用神经网络根据$d$预测$w$虽具吸引力，但训练此类模型需要协调组合优化的离散特性与神经网络训练所依赖的梯度框架。当问题涉及整数线性规划（ILP）时，克服这一训练难题的一种方法是考虑组合问题的连续松弛。尽管现有基于该方法的方案在小规模问题上表现出色，但在大规模问题上往往扩展性不足。本研究借鉴现代凸优化思想，设计了一种可轻松扩展至包含数千变量问题的网络结构与训练方案。实验验证了所提方法在两个代表性问题（最短路径问题和背包问题）上的计算优势。