Algorithmic and modeling insights via volumetric comparison of polyhedral relaxations

Jon Lee; Daphne Skipper; Emily Speakman

doi:10.1007/s10107-018-1272-6

Algorithmic and modeling insights via volumetric comparison of polyhedral relaxations

Jon Lee, Daphne Skipper, Emily Speakman

Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

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.

Original language	English (US)
Pages (from-to)	121-140
Number of pages	20
Journal	Mathematical Programming
Volume	170
Issue number	1
DOIs	https://doi.org/10.1007/s10107-018-1272-6
State	Published - Jul 1 2018

Keywords

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

ASJC Scopus subject areas

Software
Mathematics(all)

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10.1007/s10107-018-1272-6

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title = "Algorithmic and modeling insights via volumetric comparison of polyhedral relaxations",

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

author = "Jon Lee and Daphne Skipper and Emily Speakman",

note = "Funding Information: J. Lee was supported in part by ONR Grant N00014-17-1-2296. Additionally, part of this work was done while J. Lee was visiting the Simons Institute for the Theory of Computing. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF Grant #CCF-1740425. E. Speakman was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—314838170, GRK 2297 MathCoRe. Publisher Copyright: {\textcopyright} 2018, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.",

year = "2018",

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doi = "10.1007/s10107-018-1272-6",

language = "English (US)",

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AU - Skipper, Daphne

AU - Speakman, Emily

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N2 - 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.

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KW - Boolean quadric

KW - Facility location

KW - Fixed charge

KW - Global optimization

KW - Mixed-integer nonlinear optimization

KW - Monomial

KW - Polytope

KW - Spatial branch-and-bound

KW - Vertex packing

KW - Volume

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Algorithmic and modeling insights via volumetric comparison of polyhedral relaxations

Abstract

Keywords

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