Beyond Interval Uncertainty: Taking Constraints into Account

Abstract

For set information, in addition to the interval bounds on each variables x₁,..., x _n , we may have additional information: e.g., we may know that the actual values should satisfy a constraint g(x₁,..., x _n ) ≤ g₀. As we have mentioned earlier, usually, we know the approximate values of x _i , so we can safely replace the function g(x₁,..., x _n ) by, e.g., the first two terms in its Taylor expansion. In this case, the constraint becomes quadratic, and – in a realistic case when this constraint describes a bounded set – the set of all the tuples x = (x₁,..., x _n ) that satisfy this constraint forms an ellipsoid. In this case, in addition to knowing that the actual tuple x belongs to the box, we also know that it belongs to the ellipsoid – i.e., that the set of possible values of this tuple is an intersection of the box and the ellipsoid. Such a situation is analyzed in this chapter.