We consider the distributionally robust optimization and show that computing the distributional worst-case is equivalent to computing the projection onto the canonical simplex with additional linear inequality. We consider several distance functions to measure the distance of distributions. We write the projections as optimization problems and show that they are equivalent to finding a zero of real-valued functions. We prove that these functions possess nice properties such as monotonicity or convexity. We design optimization methods with guaranteed convergence and derive their theoretical complexity. We demonstrate that our methods have (almost) linear observed complexity.

我们考虑了分布稳健的优化，并表明计算分布最坏情况等效于计算投影到具有附加线性不等式的规范单纯形上。我们考虑了几个距离函数来测量分布的距离。我们将投影写为优化问题，并表明它们等价于找到实值函数的零。我们证明了这些函数具有良好的性质，例如单调性或凸性。我们设计了保证收敛的优化方法，并推导出它们的理论复杂性。我们证明我们的方法具有（几乎）线性观察到的复杂性。