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Sum of Squares Certificates for Containment of H-polytopes in V-polytopes
[article]

2016
*
arXiv
*
pre-print

Based on a formulation

arXiv:1409.5008v4
fatcat:mhd6sgv2pndlvdue6ycojl2qiq
*of*the problem*in*terms*of*a bilinear feasibility problem, we study*sum**of**squares**certificates*to decide the*containment*problem. ... Given an*H*-*polytope*P and a*V*-*polytope*Q, the decision problem whether P is*contained**in*Q is co-NP-complete. ... A preliminary version*of*this paper appeared as a regular contributed talk*for*the conference MEGA 2015 (thanks to one*of*the conference reviewers*for*providing an improved sos representation*in*Example ...##
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Sum of Squares Certificates for Containment of $\mathcal{H}$-Polytopes in $\mathcal{V}$-Polytopes

2016
*
SIAM Journal on Discrete Mathematics
*

Based on a formulation

doi:10.1137/15m1013341
fatcat:dxttdrqa5bdhxcpowfljlvcwle
*of*the problem*in*terms*of*a bilinear feasibility problem, we study*sum**of**squares**certificates*to decide the*containment*problem. ... Given an*H*-*polytope*P and a*V*-*polytope*Q, the decision problem whether P is*contained**in*Q is co-NP-complete. ... A preliminary version*of*this paper appeared as a regular contributed talk*for*the conference MEGA 2015 (thanks to one*of*the conference reviewers*for*providing an improved sos representation*in*Example ...##
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An Experimental Comparison of SONC and SOS Certificates for Unconstrained Optimization
[article]

2018
*
arXiv
*
pre-print

As a main result, we carry out the first large-scale comparison

arXiv:1808.08431v1
fatcat:rpq4gzqmg5h4vkd7qfc3opzmda
*of*SONC, using this algorithm and different geometric programming (GP) solvers, with the classical*sums**of**squares*(SOS) approach, using ... We present a polynomial time algorithm to approximate such lower bounds via*sums**of*nonnegative circuit polynomials (SONC). ... Both authors*of*this article are supported by the DFG grant WO 2206/1-1. ...##
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Sparse sums of squares on finite abelian groups and improved semidefinite lifts

2016
*
Mathematical programming
*

First,

doi:10.1007/s10107-015-0977-z
fatcat:c3kfgr6yarhdhdyjlnh3pgyylu
*in*the case where G = Z_2^n, by constructing a particular chordal cover*of*the half-cube graph, we prove that any nonnegative quadratic form*in*n binary variables is a*sum**of**squares**of*functions ... To the best*of*our knowledge, this is the first explicit family*of**polytopes**in*increasing dimensions where xc_PSD(P_d) = o(xc_LP(P_d)). ...*of*a sparse*sum*-*of*-*squares**certificate**for*any nonnegative function supported on S. ...##
###
A Semidefinite Hierarchy for Disjointly Constrained Multilinear Programming
[article]

2016
*
arXiv
*
pre-print

Under an additional geometric condition, the NP-complete

arXiv:1603.03634v1
fatcat:happcknkmzacfbb4q2c774zxky
*containment*problem*for*projections*of**H*-*polytopes*can be decided*in*finitely many steps. ...*For*nondegenerate bimatrix games, a Nash equilibrium can be computed by the*sum**of**squares*approach*in*finitely many steps. ... The proof*of*Theorem 3.1 extends the one*for*the*containment*problem*of*an*H*-*polytope**in*a*V*-*polytope**in*[11] . ...##
###
Pre- and Post-Processing Sum-of-Squares Programs in Practice

2009
*
IEEE Transactions on Automatic Control
*

This paper describes how the

doi:10.1109/tac.2009.2017144
fatcat:yufyuutivrh5rnulblzaqiklpu
*sum*-*of*-*squares*module*in*the MATLAB toolbox YALMIP handles these issues. ... Checking non-negativity*of*polynomials using*sum*-ofsquares has recently been popularized and found many applications*in*control. ... The idea*in*a*sum*-*of*-*squares*approach is to replace non-negativity with the, obviously sufficient, condition that the polynomial is a*sum**of**squared*terms. f (x) = M X i=1*h*2 i (x) = M X i=1 (q T i*v*( ...##
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Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

2015
*
SIAM Journal on Optimization
*

Our main result is a structure theorem where we show that any equivariant psd lift

doi:10.1137/140966265
fatcat:vrjpdsih7bhonjslfwpjbwpe7a
*of*size d*of*an orbitope is*of**sum*-*of*-*squares*type where the functions*in*the*sum*-*of*-*squares*decomposition come from an ... We use this framework to study two well-known families*of**polytopes*, namely the parity*polytope*and the cut*polytope*, and we prove exponential lower bounds*for*equivariant psd lifts*of*these*polytopes*. ... The authors would like to thank the anonymous referees*for*their thorough comments which helped improve the paper,*in*particular Theorem 1.6. ...##
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f-vectors of 3-polytopes symmetric under rotations and rotary reflections
[article]

2020
*
arXiv
*
pre-print

We give a full answer

arXiv:2002.00355v2
fatcat:4pzlsbd4xfbqzo7kyfaayuxo6i
*for*all three dimensional*polytopes*that are symmetric with respect to a finite rotation or rotary reflection group. ... The f-vector*of*a*polytope*consists*of*the numbers*of*its i-dimensional faces. An open field*of*study is the characterization*of*all possible f-vectors. ... Acknowledgement We like to thank Frieder Ladisch and Michael Joswig*for*many helpful advices. ...##
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Finding and Optimizing Certified, Collision-Free Regions in Configuration Space for Robot Manipulators
[article]

2022
*
arXiv
*
pre-print

Our regions are generated by alternating between two convex optimization problems: (1) a simultaneous search

arXiv:2205.03690v1
fatcat:4afsyauzpjhn5denq7igvnmjla
*for*a maximal-volume ellipse inscribed*in*a given*polytope*and a*certificate*that the*polytope*... is collision-free and (2) a maximal expansion*of*the*polytope*away from the ellipse which does not violate the*certificate*. ... Searching*for**certificates**of*non-collision*in*C-space when obstacles are specified as convex sets*in*the task space using convex programming (specifically*Sum*-*Of*-*Squares*(SOS) programming) is the primary ...##
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Equivariant Semidefinite Lifts of Regular Polygons

2017
*
Mathematics of Operations Research
*

Our construction relies on finding a sparse

doi:10.1287/moor.2016.0813
fatcat:rlzpjx525bb6jeeqhnaydc5dyq
*sum*-*of*-*squares**certificate**for*the facet-defining inequalities*of*the regular 2^n-gon, i.e., one that only uses a small (logarithmic) number*of*monomials. ... We first show that the standard Lasserre/*sum*-*of*-*squares*hierarchy*for*the regular N-gon requires exactly ceil(N/4) iterations and thus yields an equivariant psd lift*of*size linear*in*N. ... The authors would like to thank Greg Blekherman*for*giving them the permission to include his proof*of*Proposition 1. ...##
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Gram Spectrahedra
[article]

2018
*
arXiv
*
pre-print

Representations

arXiv:1608.00234v3
fatcat:vhz3ystdszg7pkx74mxo5nyxz4
*of*nonnegative polynomials as*sums**of**squares*are central to real algebraic geometry and the subject*of*active research. ... The*sum*-*of*-*squares*representations*of*a given polynomial are parametrized by the convex body*of*positive semidefinite Gram matrices, called the Gram spectrahedron. ... Lynn Chua was supported by a UC Berkeley Graduate Fellowship and the Max Planck Institute*for*Mathematics*in*the Sciences, Leipzig. ...##
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General non-realizability certificates for spheres with linear programming
[article]

2021
*
arXiv
*
pre-print

*In*this paper we present a simple technique to derive

*certificates*

*of*non-realizability

*for*an abstract

*polytopal*sphere. ... Our approach uses a variant

*of*the classical algebraic

*certificates*introduced by Bokowski and Sturmfels

*in*[Computational Synthetic Geometry, 1989], the final polynomials. ... The second author thanks Marco Macchia

*for*his suggestions on the implementation

*of*the algorithm

*in*SageMath. ...

##
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Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits
[article]

2015
*
arXiv
*
pre-print

Additionally, these statements yield a completely new class

arXiv:1402.0462v3
fatcat:7qkxa735vngpdfiac2cw7z7s7u
*of*nonnegativity*certificates*independent from*sums**of**squares**certificates*. ... We completely characterize sections*of*the cones*of*nonnegative polynomials, convex polynomials and*sums**of**squares*with polynomials supported on circuits, a genuine class*of*sparse polynomials. ... A Sufficient Condition*for**H*-simplices. By Theorem 5.2, all nonnegative polynomials*in*P y ∆ supported on an*H*-simplex are*sums**of**squares*. ...##
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Amoebas, nonnegative polynomials and sums of squares supported on circuits

2016
*
Research in the Mathematical Sciences
*

Additionally, these statements yield a completely new class

doi:10.1186/s40687-016-0052-2
fatcat:a33nmfioqvb75jnqfuhou5pmca
*of*nonnegativity*certificates*independent from*sums**of**squares**certificates*. ... We completely characterize sections*of*the cones*of*nonnegative polynomials, convex polynomials and*sums**of**squares*with polynomials supported on circuits, a genuine class*of*sparse polynomials. ... Acknowledgements We would like to thank Christian Haase*for*his support and explanations concerning toric ideals and normality. ...##
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Lower Bounds for Polynomials with Simplex Newton Polytopes Based on Geometric Programming
[article]

2016
*
arXiv
*
pre-print

We provide new sufficient conditions

arXiv:1402.6185v4
fatcat:nzp7v3qeivfcxjug4unp3ut5ui
*for*polynomials to be nonnegative as well as to have a*sum**of*binomial*squares*representation. ... Furthermore, it shows that geometric programming is strongly related to nonnegativity*certificates*based on*sums**of*nonnegative circuit polynomials, which were recently introduced by the authors. ... Acknowledgments We are deeply grateful to Alexander Kovacec*for*his detailed comments and suggestions, which significantly improved the presentation*of*the paper. ...
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