TABLE OF CONTENTS We already have met, on different occasions, with the notion of an ellipsoid-a set E in R n that can be represented as the image of the unit Euclidean ball under an affine mapping:
Ellipsoids are very convenient mathematical entities:
It is easy to specify an ellipsoid-just indicate the corresponding matrix A and vector c.
The family of ellipsoids is closed with respect to affine transformations: the image of an ellipsoid under an affine mapping again is an ellipsoid.
There are many operations, like minimization of a linear form and computation of volume that are easy to carry out when the set in question is an ellipsoid and are difficult to carry out for more general convex sets.
By the indicated reasons, ellipsoids play an important role in different areas of applied mathematics. In particular, ellipsoids are used to approximate more complicated sets. As a simple motivating example, consider a discrete time linear time invariant controlled system:
and assume that the control is norm-bounded:
The question is, What is the set X T of all states reachable in a given time T, i.e., the set of all possible values of x( T)? We can easily write down the answer:
but this answer is not explicit. To check whether a given vector x belongs to X T requires solving a nontrivial conic quadratic problem; the larger T, the greater the complexity of the problem. In fact,...
