基于混合整数规划的拍卖保留价Optimization
Contextual Reserve Price Optimization in Auctions via Mixed-Integer Programming
摘要：
我们研究的问题，学习线性模型设置拍卖底价，给定的上下文信息，以最大限度地提高卖方的预期收入。首先，我们表明，这是不可能解决这个问题，在多项式时间，除非\emph{Exponential Time Hypothesis}失败。其次，我们提出了一个强混合整数规划（MIP）制定这个问题，这是能够准确地建模的非凸和不连续的预期报酬函数。此外，我们表明，这种MIP配方是理想的（即最强的可能配方）的收入函数的一个单一的印象。由于它可以是计算昂贵的，以准确地解决MIP配方在实践中，我们还研究了其线性规划（LP）松弛的性能。虽然它可能在实践中工作得很好，我们表明，不幸的是，在最坏的情况下，LP松弛的最佳目标可以是O（样本个数）倍大于真正的问题的最佳目标。最后，我们提出了计算结果，显示MIP配方，随着其LP松弛，能够实现优越的在和出的样本性能，相比，国家的最先进的算法在真实和合成数据集。更广泛地说，我们相信这项工作提供了一个指标的优化方法，如MIP的强度，以准确地模拟机器学习问题中的内在不连续性。
英文摘要：
We study the problem of learning a linear model to set the reserve price in an auction, given contextual information, in order to maximize expected revenue from the seller side. First, we show that it is not possible to solve this problem in polynomial time unless the \emph{Exponential Time Hypothesis} fails. Second, we present a strong mixed-integer programming (MIP) formulation for this problem, which is capable of exactly modeling the nonconvex and discontinuous expected reward function. Moreover, we show that this MIP formulation is ideal (i.e. the strongest possible formulation) for the revenue function of a single impression. Since it can be computationally expensive to exactly solve the MIP formulation in practice, we also study the performance of its linear programming (LP) relaxation. Though it may work well in practice, we show that, unfortunately, in the worst case the optimal objective of the LP relaxation can be O(number of samples) times larger than the optimal objective of the true problem. Finally, we present computational results, showcasing that the MIP formulation, along with its LP relaxation, are able to achieve superior in- and out-of-sample performance, as compared to state-of-the-art algorithms on both real and synthetic datasets. More broadly, we believe this work offers an indication of the strength of optimization methodologies like MIP to exactly model intrinsic discontinuities in machine learning problems.
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