Treffer: Expansion and estimation of the range of nonlinear functions: Expansion and estimation of the range of bonlinear functions

Summary: 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.