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We consider the integer points in a unimodular cone K ordered by alexicographic rule defined by a lattice basis. To each integer point x in K weassociate a family of inequalities (lex-cuts) that defines the convex hull ofthe integer points in K that are not lexicographically smaller than x. Thefamily of lex-cuts contains the Chvatal-Gomory cuts, but does not contain andis not contained in the family of split cuts. This provides a finite cuttingplane method to solve the integer program min{cx : x \in S \cap Z^n }, where S\subset R^n is a compact set and c \in Z^n . We analyze the number ofiterations of our algorithm.

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