论文标题
从方向衍生物构建一个子级别以进行两个变量的函数
Constructing a subgradient from directional derivatives for functions of two variables
论文作者
论文摘要
对于任何是局部Lipschitz连续且方向上可区分的标量值的双变量函数，可以表明，亚级别可以始终从函数的四个指南方向上的定向衍生物构造，并以所谓的“指南针差”排列。当原始函数是非convex时，获得的亚级别是克拉克广泛梯度的元素，但是即使对于凸功能，结果也似乎是新颖的。不需要以任何特定形式表示该函数，也不需要进一步的假设，尽管在Nesterov的意义上，当函数又是L-Smooth时，结果将得到增强。对于某些最佳值函数和微分方程系统的某些参数解决方案，这些新结果似乎是计算亚级别的唯一已知方法。这些结果还表明，集中的有限差异将收敛到双变量非平滑函数的亚级别。作为双重结果，我们发现在二维设置的任何紧凑型凸都包含其间隔船体的中点。包括示例以示意图，并且证明这些结果并未直接扩展到两个以上变量或较高维度的函数。
For any scalar-valued bivariate function that is locally Lipschitz continuous and directionally differentiable, it is shown that a subgradient may always be constructed from the function's directional derivatives in the four compass directions, arranged in a so-called "compass difference". When the original function is nonconvex, the obtained subgradient is an element of Clarke's generalized gradient, but the result appears to be novel even for convex functions. The function is not required to be represented in any particular form, and no further assumptions are required, though the result is strengthened when the function is additionally L-smooth in the sense of Nesterov. For certain optimal-value functions and certain parametric solutions of differential equation systems, these new results appear to provide the only known way to compute a subgradient. These results also imply that centered finite differences will converge to a subgradient for bivariate nonsmooth functions. As a dual result, we find that any compact convex set in two dimensions contains the midpoint of its interval hull. Examples are included for illustration, and it is demonstrated that these results do not extend directly to functions of more than two variables or sets in higher dimensions.
