Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective

We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x . The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program min{cx: x∈ S∩ℤ^n} , where S⊂ℝ^n is a compact set and c∈ℤ^n . We analyze the number of iterations of our algorithm.