Polyhedra and Optimization in Connection with a Weak Majorization Ordering

We introduce the concept of weak k-majorization extending the classical notion of weak sub-majorization. For integers k and n with k n a vector x 2 R is weakly k-majorized by a vector q 2 R if the sum of the r largest components of x does not exceed the sum of the r largest components of q, for r = 1; : : : ; k. For a given q the set of vectors weakly k-majorized by q de nes a polyhedron P (q;k), and we determine all its vertices. We also determine the vertices and a complete and nonredundant linear description of the integer hull of P (q;k). The results are used to give simple and e cient (polynomial time) algorithms for associated linear and integer linear programming problems.