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∈ℝn is weakly k-majorized by a vector q∈ℝk 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 defines 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 efficient (polynomial time) algorithms for associated linear and integer linear programming problems.
KeywordsMajorization polyhedral combinatorics
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