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

Michele Conforti (), Marianna Santis (), Marco Summa () and Francesco Rinaldi ()
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Michele Conforti: Università degli Studi di Padova
Marianna Santis: Sapienza Università di Roma
Marco Summa: Università degli Studi di Padova
Francesco Rinaldi: Università degli Studi di Padova

4OR, 2021, vol. 19, issue 4, No 3, 548 pages

Abstract: 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 $$\min \{cx: x\in S\cap \mathbb {Z}^n\}$$ min { c x : x ∈ S ∩ Z n } , where $$S\subset \mathbb {R}^n$$ S ⊂ R n is a compact set and $$c\in \mathbb {Z}^n$$ c ∈ Z n . We analyze the number of iterations of our algorithm.

Keywords: Nonlinear integer programming; Valid inequalities; Cutting plane method; 90C10; 90C57 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10288-020-00459-6

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