We study an online revenue maximization problem where the consumers arrive i.i.d from some unknown distribution and purchase a bundle of products from the sellers. The classical approach generally assumes complete knowledge of the consumer utility functions, while recent works have been devoted to unknown linear utility functions. This paper focuses on the online posted-price model with unknown consumer distribution and unknown consumer utilities, given they are concave. Hence, the two questions to ask are i) when is the seller's online maximization problem concave, and ii) how to find the optimal pricing strategy for non-linear utilities. We answer the first question by imposing a third-order smoothness condition on the utilities. The second question is addressed by two algorithms, which we prove to exhibit the sub-linear regrets of $O(T^{\frac{2}{3}} (\log T)^{\frac{1}{3}})$ and $O(T^{\frac{1}{2}} (\log T)^{\frac{1}{2}})$ respectively.
翻译：我们研究的是消费者抵达的网上收入最大化问题,即消费者来自一些未知的销售品并从销售商那里购买一捆产品。古典方法一般假定完全了解消费者公用事业功能,而最近的作品则专门用于未知的线性公用事业功能。本文侧重于在线上公布的价格模式,消费者分布不明,消费公用事业也不为人所知。因此,要问的两个问题是:(一)卖方的网上最大化问题何时出现;(二)如何找到非线性公用事业的最佳定价战略。我们通过对公用事业实行三等平滑条件回答第一个问题。第二个问题由两种算法解决,我们证明这些算法分别显示了美元(T ⁇ frac{2 ⁇ 3 ⁇ 3 ⁇ 3 ⁇ 3美元)和美元(T ⁇ frac{1 ⁇ 2 ⁇ 2 ⁇ 2美元)的亚线性遗憾。
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