用KKT条件训练神经网络，直接求解凸优化问题。
Karush-Kuhn-Tucker Condition-Trained Neural Networks (KKT Nets)
- 以KKT条件为损失函数，让网络自动生成最优解
- 仅优化KKT损失比混合损失效果更好，且优于单纯误差最小化
- 适合需要快速求解凸优化的场景，如实时控制与资源分配
本文提出一种新方法，通过利用在特定正则条件下，满足Karush-Kuhn-Tucker（KKT）条件的原始变量或对偶变量即为最优解这一性质，来求解凸优化问题。类似于理论训练神经网络（TTNNs），将优化问题的参数输入网络，期望输出为最优原始变量和对偶变量。采用一种称为KKT损失的损失函数，衡量网络输出满足KKT条件的程度。以线性规划为例，实验表明：仅最小化KKT损失的表现优于同时最小化KKT损失与数据损失（即网络输出与真实最优解的均方误差）；而仅最小化数据损失的效果更差。尽管该方法有潜力，但得到的原始和对偶解仍不够接近真实最优解。未来工作将致力于改进模型，使解更精确，并拓展到其他问题类别。
原文摘要 · Abstract (English)
This paper presents a novel approach to solving convex optimization problems by leveraging the fact that, under certain regularity conditions, any set of primal or dual variables satisfying the Karush-Kuhn-Tucker (KKT) conditions is necessary and sufficient for optimality. Similar to Theory-Trained Neural Networks (TTNNs), the parameters of the convex optimization problem are input to the neural network, and the expected outputs are the optimal primal and dual variables. A choice for the loss function in this case is a loss, which we refer to as the KKT Loss, that measures how well the network's outputs satisfy the KKT conditions. We demonstrate the effectiveness of this approach using a linear program as an example. For this problem, we observe that minimizing the KKT Loss alone outperforms training the network with a weighted sum of the KKT Loss and a Data Loss (the mean-squared error between the ground truth optimal solutions and the network's output). Moreover, minimizing only the Data Loss yields inferior results compared to those obtained by minimizing the KKT Loss. While the approach is promising, the obtained primal and dual solutions are not sufficiently close to the ground truth optimal solutions. In the future, we aim to develop improved models to obtain solutions closer to the ground truth and extend the approach to other problem classes.
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