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A Representation of the Interval Hull of a Tolerance Polyhedron Describing Inclusions of Function Values and Slopes

Authors:

Heindl Gerhard

Affiliation:

(1) Fachbereich Mathematik, Universität Wuppertal, Gaußstraße 20, 42097 Wuppertal, Germany

Abstract:

Given a nonempty set of functions

$\begin{gathered} F = \{ f:a,b] \to \mathbb{R}: \hfill \\ \hfill \\ {\text{ }}f(x_i ) \in w_i ,i = 0, \ldots ,n,{\text{ and}} \hfill \\ \hfill \\ {\text{ }}f(x) - f(y) \in d_i (x - y){\text{ }}\forall x,y \in x_{i - 1} ,x_i ],{\text{ }}i = 1, \ldots ,n\} , \hfill \\ \end{gathered}$

where a = x ₀ < ... < x _n = b are known nodes and w _i, i = 0,...,n, d _i, i = 1,..., n, known compact intervals, the main aim of the present paper is to show that the functions $\underline f :x \mapsto \min \{ f(x):f \in F\} ,{\text{ }}x \in a,b],$ and

$\overline f :x \mapsto \max \{ f(x):f \in F\} ,{\text{ }}x \in a,b],$

exist, are in F, and are easily computable. This is achieved essentially by giving simple formulas for computing two vectors $\tilde l,\tilde u \in \mathbb{R}^{n + 1}$ with the properties

$\begin{gathered} \bullet {\text{ }}\tilde l \leqslant \tilde u{\text{ implies}} \hfill \\ \hfill \\ {\text{ }}\tilde l,\tilde u \in T{\text{ : = \{ }}\xi {\text{ = (}}\xi _0 , \ldots ,\xi _n )^T \in \mathbb{R}^{n + 1} : \hfill \\ \hfill \\ {\text{ }}\xi _i \in w_i ,{\text{ }}i = 0, \ldots ,n,{\text{ and}} \hfill \\ \end{gathered}$