Slope Interval, Generalized Gradient, Semigradient, and Slant Derivative

Many practical optimization problems are nonsmooth, and derivative-type methods cannot be applied. To overcome this difficulty, there are different approaches to replace the derivative of a function f : Rn → R: interval slopes, semigradients, generalized gradients, and slant derivatives are some examples. 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. Illustrative examples are shown in both cases.