The notion of optimality is not as simple for linear interval arithmetic as it was for constant interval arithmetic. Consider the following function g, with two distinct bounds, and :
A bound is optimal, for linear interval arithmetic, if no better linear interval upper bound exists:
Also, arguing as before, we may show that an optimal model of g is an interval extension of g. This implies that for differentiable g, an optimal model produces bounds which touch g at two distinct points, allowing for infinitesimal separation between points. Infinitesimally separated points correspond to the upper bound matching both the value and the derivative at a point.
Jeff Tupper | March 1996 |