Abstract
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.
Highlights
For any scalar-valued bivariate function that is locally Lipschitz continuous and directionally di erentiable, 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 di erence”
For certain optimal-value functions and certain parametric solutions of di erential equation systems, these new results appear to provide the only known way to compute a subgradient. These results imply that centered nite di erences will converge to a subgradient for bivariate nonsmooth functions
We nd that any compact convex set in two dimensions contains the midpoint of its interval hull
Summary
For any scalar-valued bivariate function that is locally Lipschitz continuous and directionally di erentiable, 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 di erence”. Danskin’s classical result [ , ] describes directional derivatives for optimal-value functions as the solutions of related optimization problems in a general setting, while subgradient results such as [ , Theorem . Directional derivatives and subdi erentials of convex functions are essentially duals [ ].
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