Convexification of bilinear terms over network polytopes
Date
2023-02-27
Authors
Khademnia, Erfan
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arXiv
Abstract
It is well-known that the McCormick relaxation for the bilinear constraint z=xy gives the convex hull over the box domains for x and y. In network applications where the domain of bilinear variables is described by a network polytope, the McCormick relaxation, also referred to as linearization, fails to provide the convex hull and often leads to poor dual bounds. We study the convex hull of the set containing bilinear constraints zi,j=xiyj where xi represents the arc-flow variable in a network polytope, and yj is in a simplex. For the case where the simplex contains a single y variable, we introduce a systematic procedure to obtain the convex hull of the above set in the original space of variables, and show that all facet-defining inequalities of the convex hull can be obtained explicitly through identifying a special tree structure in the underlying network. For the generalization where the simplex contains multiple y variables, we design a constructive procedure to obtain an important class of facet-defining inequalities for the convex hull of the underlying bilinear set that is characterized by a special forest structure in the underlying network. Computational experiments are presented to evaluate the effectiveness of the proposed methods.
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Preprint
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This is a pre-print of the article Khademnia, Erfan, and Danial Davarnia. "Convexification of bilinear terms over network polytopes." arXiv preprint arXiv:2302.14151 (2023).
DOI: 10.48550/arXiv.2302.14151.
Attribution 4.0 International (CC BY 4.0).
Copyright 2023 The Authors.
Posted with permission.