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

Given a nonempty set of functions $$\begin{gathered} F = \{ f:[a,b] \to \mathbb{R}: \hfill \\ f({x_i}) \in {w_i}, i = 0,...,n, and \hfill \\ f(x) - f(y) \in {d_i}(x - y) \forall x,y \in [{x_{i - 1}},{x_i}], i = 1,...,n\} ,\hfill \\ \end{gathered} $$ 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 $$\begin{gathered} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} :x \mapsto \min \{ f(x):f \in F\} ,x \in [a,b], and \hfill \\ f:x \mapsto \max \{ f(x):f \in F\} ,x \in [a,b], \hfill \\ \end{gathered} $$ 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 \(\tilde l \leqslant \tilde u\) implies $$\begin{gathered} \tilde l, \tilde u \in T: = \{ \xi = {({\xi _0},...,{\xi _n})^T} \in {\mathbb{R}^{n + 1}}: \hfill \\ {\xi _i} \in {w_i}, i = 0,...,n, and \hfill \\ {\xi _i} - {\xi _{i - 1}} \in {d_i}({x_i} - {x_{i - 1}}),i = 1,...,n\} \hfill \\ \end{gathered} $$ and that \([\tilde l,\tilde u]\) is the interval hull of (the tolerance polyhedron) T; \(\tilde l \leqslant \tilde u\) iff T ≠ o iff F ≠ o. \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} ,\bar f\) can serve for solving the following problem: