New algorithms for constructing optimal circumscribed and inscribed ellipsoids

This paper is an overview of some methods used to solve the problem of constructing an ellipsoid of minimal volume containing a set of m points and the problem of constructing an ellipsoid of maximal volume inscribed in a polyhedron defined by a system of m linear inequalities in n -dimensional Euclidean space (using nonsmooth convex penalty functions and successive space transformation). The type 1 algorithms require O( n 3) or O(m 3) arithmetical operations per iteration (depending on whether the prime or dual algorithm is considered); the type 2 algorithms, require O{nm) arithmetical operations per iteration.