Minimal Bounding Ellipsoid
Given a set of 
points 
in 
dimensions, find the Löwner–John ellipsoid that is the smallest-volume ellipsoid enclosing all of the points.
This example demonstrates that for problems expressed in terms of highly nonlinear convex objective functions and constraints, NMinimize can do transformations that make the problem easily solved by convex optimization.
An ellipsoid in 
can be described by the set of points 
such that 
, where 
is a positive semidefinite 
matrix and 
is an 
-vector. The volume of the ellipsoid is proportional to 
. 
is convex so finding the solution of
The example shown here is done for a random set of points in three dimensions so that it can be visualized, but the solution algorithm can handle larger dimensions 
.
Find the solution using NMinimize.
In three dimensions, the ellipsoid can be described in terms of the Ellipsoid graphics primitive.
BoundingRegion can find an enclosing ellipsoid faster, but it has greater volume.
