next up previous notation contents
Next: 3.3.4 Piecewise Models Up: 3.3 Linear Interval Arithmetic Previous: 3.3.2 Charts

3.3.3 Optimality

 

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, tex2html_wrap_inline35677 and tex2html_wrap_inline35679 :

figure17443

We now define a measure of bound goodness. Consider the tex2html_wrap_inline31325 norm:    

math17539

where tex2html_wrap_inline35695 is a continuous positive function defined on [0,1]. The tex2html_wrap_inline31325 norm is always positive, since we consider only tex2html_wrap_inline35701 which are upper bounds of g. We consider the upper bound tex2html_wrap_inline35705 to be a better upper bound than tex2html_wrap_inline35679 if tex2html_wrap_inline35709 : a bound tex2html_wrap_inline35701 is good if tex2html_wrap_inline35713 is small.

A bound tex2html_wrap_inline35715 is optimal, for linear interval arithmetic, if no better linear interval upper bound exists:

math17553

The model tex2html_wrap_inline35717 returns optimal upper bounds if the upper bound is optimal for all tex2html_wrap_inline32905 :

math17562

As before, optimality can be defined without reference to the underlying function.

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.


next up previous notation contents
Next: 3.3.4 Piecewise Models Up: 3.3 Linear Interval Arithmetic Previous: 3.3.2 Charts
Jeff TupperMarch 1996