An iterative algorithm for finding a nearest pair of points in two convex subsets of Rn

We present an algorithm for finding a nearest pair of points in two convex sets of Rn, and therefore, their distance. The algorithm is based on the fixed-point theory of nonexpansive operators on a Hilbert space. Its practical implementation requires a fast projection algorithm. We introduce such a procedure for convex polyhedra. This algorithm effects a local search in the faces using visibility as a guide for finding the global minimum. After studying the convergence of both algorithms, we detail computer experiments on polyhedra (projection and distance). In the case of distances, these experiments show a sublinear time complexity relative to the total number of vertices.