Quantifying Double McCormick

• 成果类型: Article
• 署名作者: Speakman, Emily; Lee, Jon
• 署名单位: University of Michigan System; University of Michigan
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2017.0846
• 发表日期: 2017
• 页码: 1230-1253
• 关键词: global optimization; Convex envelope; bounds

摘要

When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the bounds defining the box. In this manner, we precisely quantify the strength of the different possible relaxations defined by all three groupings, in addition to the trilinear hull itself. As a by-product, we characterize the best double-McCormick relaxation. We wish to emphasize that, in the context of spatial branch and bound for factorable formulations, our results do not only apply to variables in the input formulation. Our results apply to monomials that involve auxiliary variables as well. So, our results apply to the product of any three (possibly complicated) expressions in a formulation.