Slope Intervals, Generalized Gradients, Semigradients, Slant Derivatives, and Csets

Abstract

Many practical optimization problems are nonsmooth, and derivative-type methods cannot be applied. To overcome this difficulty, there are different concepts to replace the derivative of a function f : $\mathbb{R}^n \to \mathbb{R}$: interval slopes, semigradients, generalized gradients, and slant derivatives are some examples. These approaches generalize the success of convex analysis, and are effective in optimization. However, with the exception of interval slopes, it is not clear how to automatically compute these; having a general analogue to the chain rule, interval slopes can be computed with automatic differentiation techniques. In this paper we study the relationships among these approaches for nonsmooth Lipschitz optimization problems in finite dimensional Euclidean spaces. Inclusion theorems concerning the equivalence of these concepts when there exist one sided derivatives in one dimension and in multidimensional cases are proved separately. Valid enclosures are produced. Under containment set (cset) theory, for instance, the cset of the gradient of a locally Lipschitz function f near x is included in its generalized gradient.