We study the nonlinear newsvendor problem concerning goods of a non-discrete nature, and a class of stochastic dynamic programs with several application areas such as supply chain management and economics. The class is characterized by continuous state and action spaces, either convex or monotone cost functions that are accessed via value oracles, and affine transition functions. We establish that these problems cannot be approximated to any degree of either relative or additive error, regardless of the scheme used. To circumvent these hardness results, we generalize the concept of fully polynomial-time approximation scheme allowing arbitrarily small additive and multiplicative error at the same time, while requiring a polynomial running time in the input size and the error parameters. We develop approximation schemes of this type for the classes of problems mentioned above. In light of our hardness results, such approximation schemes are “best possible”. A computational evaluation shows the promise of this approach.

我们研究了关于非离散货物的非线性报童问题，以及一类具有多个应用领域（如供应链管理和经济学）的随机动态程序。该类的特点是连续的状态和动作空间，通过值预言机访问的凸或单调成本函数，以及仿射转换函数。我们确定，无论使用何种方案，这些问题都不能近似到任何程度的相对或附加误差。为了规避这些硬度结果，我们概括了完全多项式时间近似方案的概念，允许同时允许任意小的加法和乘法误差，同时需要输入大小和误差参数的多项式运行时间。我们为上述问题类别开发了这种类型的近似方案。根据我们的硬度结果，这种近似方案是“最好的”。计算评估显示了这种方法的前景。