Unique lifting of integer variables in minimal inequalities

• 成果类型: Article
• 署名作者: Basu, Amitabh; Campelo, Manoel; Conforti, Michele; Cornuejols, Gerard; Zambelli, Giacomo
• 署名单位: University of California System; University of California Davis; Universidade Federal do Ceara; University of Padua; Carnegie Mellon University; University of London; London School Economics & Political Science
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-012-0560-9
• 发表日期: 2013
• 页码: 561-576
• 关键词: programs; cuts

摘要

This paper contributes to the theory of cutting planes for mixed integer linear programs (MILPs). Minimal valid inequalities are well understood for a relaxation of an MILP in tableau form where all the nonbasic variables are continuous; they are derived using the gauge function of maximal lattice-free convex sets. In this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities. We characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting (This result first appeared in the proceedings of IPCO 2010). The answer is a nonconvex region that can be obtained as a finite union of convex polyhedra. We then establish a necessary and sufficient condition for the uniqueness of the lifting function.