Expansion and Estimation of the Range of Nonlinear Functions

Many verification algorithms use an expansion $f(x) \in f(\tilde {x}) + S \cdot (x - \tilde {x})$, $f : \mathbb {R} ^n \rightarrow \mathbb {R} ^n$ for $x \in X$, where the set of matrices $S$ is usually computed as a gradient or by means of slopes. In the following, an expansion scheme is described which frequently yields sharper inclusions for $S$. This allows also to compute sharper inclusions for the range of $f$ over a domain. Roughly speaking, $f$ has to be given by means of a computer program. The process of expanding $f$ can then be fully automatized. The function $f$ need not be differentiable. For locally convex or concave functions special improvements are described. Moreover, in contrast to other methods, $\tilde {x} \cap X$ may be empty without implying large overestimations for $S$. This may be advantageous in practical applications.