Projection onto polyhedra in outer representation

The projection of the origin onto an $n$-dimensional polyhedron defined by a system of $m$ inequalities is reduced to a sequence of projection problems onto a one-parameter family of shifts of a polyhedron with at most $m+1$ vertices in $n+1$ dimensions. The problem under study is transformed into the projection onto a convex polyhedral cone with m extreme rays, which considerably simplifies the solution to an equivalent problem and reduces it to a single projection operation. Numerical results obtained for random polyhedra of high dimensions are presented.