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

Abstract

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 ≠ ø iff F ≠ ø.

$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} ,\bar f$ can serve for solving the following problem:

Assume that μ is a monotonically increasing functional on the set of Lipschitz-continuous functions f : [a, b] → ℝ (e.g. μ(f) = ſ ^b _a f (x) dx or μ(f)= min f ([a, b]) or μ (f) = max f ([a, b])), and that the available information about a function g : [a, b] →ℝ is “g ∈ F,” then the problem is to find the best possible interval inclusion of μ (g). Obviously, this inclusion is given by the interval $[\mu (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} ),\mu (\bar f)].$ Complete formulas for computing this interval are given for the case μ (f) = ſ ^b _a f(x) dx.