The MDM Algorithm and the Sylvester Problem

Abstract

When developing numerical methods for solving nonlinear minimax problems, the following auxiliary problem arose: in the convex hull of a certain finite set in Euclidean space, find a point that has the smallest norm. In 1971, B. Mitchell, V. Demyanov and V. Malozemov proposed a non-standard algorithm for solving this problem, which was later called the MDM algorithm (based on the first letters of the authors' last names). This article considers a specific minimax problem: finding the smallest volume ball containing a given finite set of points. It is called the Sylvester problem and is a special case of the problem about the Chebyshev center of a set. The Sylvester problem is associated with a convex quadratic programming problem with simplex constraints. To solve this problem, it is proposed to use a variant of the MDM algorithm. With its help, a minimizing sequence of feasible solutions is constructed such that two consecutive feasible solutions differ in only two components. The indices of these components are selected based on certain optimality conditions. We prove the weak convergence of the resulting sequence of feasible solutions that implies that the corresponding sequence of vectors converges in norm to a unique solution to the Sylvester problem. Four typical examples on a plane are given.