Outer Box Approximations

Turning to the outer approximation problem, we first consider the problem of minimizing the size of a box subject to the condition that it contains the intersection. In 2D or 3D, we again encounter an easy problem, as it suffices to find how far we can go in along coordinate axes, in both positive and negative direction along these axes. In 2D, this amounts to solve 4 problems, and in 3D, 8 problems, of the form

where stands for the -th unit vector (with zero elements, except a in the -th spot).

This gives us a box . Our estimated position is then simply the center of the box.

Outer box approximation to the intersection. This provides an estimated point (the center of the outer box), with a pessimistic estimate of the uncertainty around it. Here, the uncertainty is given as two intervals of confidence on each of the coordinates of the estimated point.