Interior Point Solving for LP-based prediction+optimisation
Interior Point Solving for LP-based prediction+optimisation
复制标题
基于 LP 的预测优化的内点求解
DOI：
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发表时间：
2020
期刊：
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通讯作者：
Tias Guns
中科院分区：
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作者：
Jayanta Mandi;Tias Guns
在许多实际分析应用中，解决优化问题是决策的关键。然而，优化问题的系数往往是不确定的，并且依赖于外部因素，如未来的需求或能源或股票价格。机器学习(ML)模型，特别是神经网络，正越来越多地被用于以数据驱动的方式估计这些系数。因此，端到端的预测和优化方法受到了越来越多的关注，这种方法考虑了预测值对解决优化问题的有效性。对于整数线性规划问题，克服其不可微性的一种流行方法是在连续松弛的基础上增加一个二次惩罚项，这样就可以使用对二次规划进行微分的结果。相反，我们研究了更有原则的对数障碍项的使用，它被广泛用于线性规划的内点求解器中。具体地说，我们不再区分KKT条件，而是考虑线性规划的齐次自对偶形式，并证明了内点步长方向与学习所需的相应梯度之间的关系。最后，我们的实证实验表明，我们的方法的性能与Wilder等人提出的最先进的QPTL(二次规划任务损失)公式一样好，甚至更好。Elmachoub和Grigas的SPO方法。
Solving optimization problems is the key to decision making in many real-life analytics applications. However, the coefficients of the optimization problems are often uncertain and dependent on external factors, such as future demand or energy or stock prices. Machine learning (ML) models, especially neural networks, are increasingly being used to estimate these coefficients in a data-driven way. Hence, end-to-end predict-and-optimize approaches, which consider how effective the predicted values are to solve the optimization problem, have received increasing attention. In case of integer linear programming problems, a popular approach to overcome their non-differentiabilty is to add a quadratic penalty term to the continuous relaxation, such that results from differentiating over quadratic programs can be used. Instead we investigate the use of the more principled logarithmic barrier term, as widely used in interior point solvers for linear programming. Specifically, instead of differentiating the KKT conditions, we consider the homogeneous self-dual formulation of the LP and we show the relation between the interior point step direction and corresponding gradients needed for learning. Finally our empirical experiments demonstrate our approach performs as good as if not better than the state-of-the-art QPTL (Quadratic Programming task loss) formulation of Wilder et al. and SPO approach of Elmachtoub and Grigas.