  We study the problem of finding the Lowner-John ellipsoid, i.e., an ellipsoid
with minimum volume that contains a given convex set. We reformulate the
problem as a generalized copositive program, and use that reformulation to
derive tractable semidefinite programming approximations for instances where
the set is defined by affine and quadratic inequalities. We prove that, when
the underlying set is a polytope, our method never provides an ellipsoid of
higher volume than the one obtained by scaling the maximum volume inscribed
ellipsoid. We empirically demonstrate that our proposed method generates
high-quality solutions faster than solving the problem to optimality.
Furthermore, we outperform the existing approximation schemes in terms of
solution time and quality. We present applications of our method to obtain
piecewise-linear decision rule approximations for dynamic distributionally
robust problems with random recourse, and to generate ellipsoidal
approximations for the set of reachable states in a linear dynamical system
when the set of allowed controls is a polytope.
