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Convexification of Bilinear Terms over Network Polytopes

Author

Listed:
  • Erfan Khademnia

    (Industrial and Manufacturing Systems Engineering, Iowa State University, Ames, Iowa 50011)

  • Danial Davarnia

    (Industrial and Manufacturing Systems Engineering, Iowa State University, Ames, Iowa 50011)

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 z i , j = x i y j where x i represents the arc-flow variable in a network polytope, and y j 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 conducted on different applications show the effectiveness of the proposed methods in improving the dual bounds obtained from alternative techniques.

Suggested Citation

  • Erfan Khademnia & Danial Davarnia, 2025. "Convexification of Bilinear Terms over Network Polytopes," Mathematics of Operations Research, INFORMS, vol. 50(2), pages 1019-1041, May.
  • Handle: RePEc:inm:ormoor:v:50:y:2025:i:2:p:1019-1041
    DOI: 10.1287/moor.2023.0001
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