Algorithmic and modeling insights via volumetric comparison of polyhedral relaxations

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Abstract

This is mostly a survey on some mathematical results concerning volumes of polytopes of interest in non-convex optimization. Our motivation is in geometrically comparing relaxations in the context of mixed-integer linear and nonlinear optimization, with the goal of gaining algorithmic and modeling insights. We consider relaxations of: fixed-charge formulations, vertex packing polytopes, boolean-quadric polytopes, and relaxations of graphs of monomials on box domains. Besides surveying the area, we do give a few new results, and we provide many directions for further work.

Keywords

Polytope Volume Global optimization Mixed-integer nonlinear optimization Fixed charge Facility location Vertex packing Boolean quadric Monomial Spatial branch-and-bound 

Mathematics Subject Classification

52B11 52B12 90C10 90C11 90C26 90C27 90C57 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of IOEUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsU.S. Naval AcademyAnnapolisUSA
  3. 3.Institute of Mathematical OptimizationOtto-von-Guericke-UniversitätMagdeburgGermany

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