Journal
MATHEMATICAL PROGRAMMING
Volume 170, Issue 1, Pages 121-140Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s10107-018-1272-6
Keywords
Polytope; Volume; Global optimization; Mixed-integer nonlinear optimization; Fixed charge; Facility location; Vertex packing; Boolean quadric; Monomial; Spatial branch-and-bound
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Funding
- ONR [N00014-17-1-2296]
- DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF [CCF-1740425]
- Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [314838170, GRK 2297 MathCoRe]
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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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