A dual algorithm for finding the minimum-norm point in a polytope

The authors give a dual algorithm for the problem of finding the minimum-norm point in the convex hull of a given finite set of points in a Euclidean space. The present algorithm repeatedly rotates a separating supporting-hyperplane and in finitely many steps finds the farthest separating supporting-hyperplane, whose minimum-norm point is the desired minimum-norm point in the polytope. During the execution of the algorithm the distance of the separating supporting-hyperplane monotonically increases. The algorithm is closely related to P. Wolfe’s primal algorithm which finds a sequence of norm-decreasing points in the given polytope. Computational experiments are carried out to show the behavior of the present algorithm.