构造二元函数的次梯度：由方向导数出发
Constructing a subgradient from directional derivatives for functions of two variables
AI总结：
本文证明对局部Lipschitz且方向可微的二元函数，可由四个罗盘方向导数构造次梯度，适用于非凸与凸情形，并给出对偶几何结果及示例，但不适用于高维。
摘要：
对于任意标量值的二元函数，若其局部Lipschitz连续且方向可微，则证明总可以依据该函数在四个罗盘方向上的方向导数构造出一个次梯度，这些方向导数按所谓的“罗盘差分”排列。当原函数非凸时，所得次梯度属于Clarke广义梯度的元素，但即使对于凸函数，该结果似乎也是新颖的。该函数无需以任何特定形式表示，也无需额外假设，尽管当函数在Nesterov意义下额外具有L-光滑性时，结果会得到加强。对于某些最优值函数和某些微分方程组的参数解，这些新结果似乎提供了计算次梯度的唯一已知途径。这些结果还意味着中心有限差分将收敛到二元非光滑函数的次梯度。作为对偶结果，我们发现二维中的任何紧凸集都包含其区间包络的中点。文中包含示例以作说明，并证明这些结果不能直接推广到多于两个变量的函数或更高维的集合。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
