3.2.5 Optimality

Consider the interval model of the function . The function g has many interval models; we will now define when the model is optimal.

A bound is optimal, for interval arithmetic, if no better upper bound exists:

The model returns optimal upper bounds if the upper bound is optimal for all :

We may now prove that the interval extension of g is optimal. Consider the upper bound, for argument j:

from the definition of interval extension. The only way could fail to be optimal is for there to be a better bound of g(j). This contradicts the definition of supremum; let the better bound be denoted as l,

or, equivalently:

but:

We now know that if g is differentiable over , then the upper bound given by is obtained by for some in j:

since j is closed. Lower bounds are handled similarly, and will be addressed in section .

Optimality can be defined without direct reference to the underlying function:

It is clear that since if then is clearly not optimal, so is not optimal. If is valid, then .

Jeff Tupper

March 1996