Abstract
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: xin Scap mathbb {Z}^n}, where Ssubset mathbb {R}^n is a compact set and cin mathbb {Z}^n. We analyze the number of iterations of our algorithm.
Highlights
The area of integer nonlinear programming is rich in applications but quite challenging from a computational point of view
To the best of our knowledge, these results are obtained under some restrictive conditions: Typically, the feasible set is assumed to be convex or to contain 0/1 points only
The lex-inequalities that we introduce in this paper are defined for a given lattice basis of Zn
Summary
The area of integer nonlinear programming is rich in applications but quite challenging from a computational point of view. While in all the papers cited above the correctness of the algorithms is based on a specific procedure for solving the continuous relaxation, there are methods that only assume that an optimal solution of the continuous relaxation is given by a black box This is the case for the lift-and-project method of Balas et al (1993) for mixed 0/1 linear problems, the procedure described by Orlin (1985) for 0/1 integer linear programming, and the algorithm presented by Neto (2012) for integer linear programming over bounded sets. We notice that a common feature of the above papers is the (explicit or implicit) use of some lexicographic rule for the choice of an optimal solution of the continuous relaxation or the selection of the cut This seems to be a key tool to prove finite convergence of this type of algorithms.
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