On the elimination of inessential points in the smallest enclosing ball problem. (4th March 2019) - Pronzato, Luc
Abstract : We consider the construction of the smallest ballB ∗ enclosing a setX n formed by n points inR d . We show that any probability measure onX n, with mean c and variance matrix V, provides a lower bound b on the distance to c of any point on the boundary ofB ∗, with b having a simple expression in terms of c and V . This inequality permits to remove inessential points fromX n, which do not participate to the definition ofB ∗, and can be used to accelerate algorithms for the construction ofB ∗ . We show that this inequality is, in some sense, the best possible. A series of numerical examples indicates that, when d is reasonably small (d ≤ 10, say) and n is large (up to10 5 ), the elimination of inessential points by a suitable two-point measure, followed by a direct (exact) solution by quadratic programming, outperforms iterative methods that compute an approximate solution by solving the dual problem.