Realizability and inscribability for simplicial polytopes via nonlinear optimization

We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. In order to show non-realizability of simplicial spheres, we extend the method of finding biquadratic final polynomials for matroid polytopes to partial mat...

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Bibliographic Details
Published in:Mathematical programming Vol. 166; no. 1-2; pp. 273 - 295
Main Author: Firsching, Moritz
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2017
Springer Nature B.V
Subjects:
Calculus of Variations and Optimal Control; Optimization
Classification
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Nonlinear programming
Numerical Analysis
Optimization
Optimization techniques
Polytopes
Realizability
Theoretical
52C40
52B40
90C30
52B11
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. In order to show non-realizability of simplicial spheres, we extend the method of finding biquadratic final polynomials for matroid polytopes to partial matroid polytopes. Combining these two methods we obtain a complete classification of neighborly polytopes of dimension 4, 6 and 7 with 11 vertices, of neighborly 5-polytopes with 10 vertices, as well as a complete classification of simplicial 3-spheres with 10 vertices into polytopal and non-polytopal spheres. Surprisingly many of the realizable polytopes are also inscribable.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-017-1120-0