Constructing a Subgradient from Directional Derivatives for Functions of Two Variables

For any scalar-valued bivariate function that is locally Lipschitz continuousand directionally differentiable, it is shown that a subgradient may always beconstructed from the function's directional derivatives in the four compassdirections, arranged in a so-called "compass difference". When the originalfunction is nonconvex, the obtained subgradient is an element of Clarke'sgeneralized gradient, but the result appears to be novel even for convexfunctions. The function is not required to be represented in any particularform, and no further assumptions are required, though the result isstrengthened when the function is additionally L-smooth in the sense ofNesterov. For certain optimal-value functions and certain parametric solutionsof differential equation systems, these new results appear to provide the onlyknown way to compute a subgradient. These results also imply that centeredfinite differences will converge to a subgradient for bivariate nonsmoothfunctions. As a dual result, we find that any compact convex set in twodimensions contains the midpoint of its interval hull. Examples are includedfor illustration, and it is demonstrated that these results do not extenddirectly to functions of more than two variables or sets in higher dimensions.