An Algorithm for Solving the Minimum-Norm Point Problem over the Intersection of a Polytope and an Affine Set

In this paper, we propose an efficient algorithm for finding the minimum-norm point in the intersection of a polytope and an affine set in an n-dimensional Euclidean space, where the polytope is expressed as the convex hull of finitely many points and the affine set is expressed as the intersection of k hyperplanes, k≥1. Our algorithm solves the problem by using directly the original points and the hyperplanes, rather than treating the problem as a special case of the general quadratic programming problem. One of the advantages of our approach is that our algorithm works as well for a class of problems with a large number (possibly exponential or factorial in n) of given points if every linear optimization problem over the convex hull of the given points is solved efficiently. The problem considered here is highly degenerate, and we take care of the degeneracy by solving a subproblem that is a conical version of the minimum-norm point problem, where points are replaced by rays. When the number k of hyperplanes expressing the affine set is equal to one, we can easily avoid degeneracy, but this is not the case for k≥2. We give a subprocedure for treating the degenerate case. The subprocedure is interesting in its own right. We also show the practical efficiency of our algorithm by computational experiments.