Test for intersection of two N-dimensional ellipsoids

Question

Let's say I have two $N$-dimensional ellipsoids:

$$ \sum_{i=1}^{N} \frac{(x_i - b_i)^2}{c_i^2} = 1 $$ $$ \sum_{i=1}^{N} \frac{(x_i - b'_i)^2}{c_i'^2} = 1 $$

How can I tell if the two intersect? Is there a computationally easy way to do this test?

Answer 1

I think you can reduce this to a convex optimization problem. The problem is a feasibility problem in which you ask if there is a point $x$ such that $(x-x_a)^T A (x-x_a) \le 1$ and $(x-x_b)^T B (x-x_b) \le 1$, where $A$ and $B$ are the quadratic forms related to the ellipsoids, and $x_a$ and $x_b$ are their centers. These types of problems can be solved efficiently, but typically require a large programming library for implementing them.