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

Kai Kellner, Thorsten Theobald
2016 arXiv   pre-print
Based on a formulation 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  ... 
arXiv:1409.5008v4 fatcat:mhd6sgv2pndlvdue6ycojl2qiq

Sum of Squares Certificates for Containment of $\mathcal{H}$-Polytopes in $\mathcal{V}$-Polytopes

Kai Kellner, Thorsten Theobald
2016 SIAM Journal on Discrete Mathematics  
Based on a formulation 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  ... 
doi:10.1137/15m1013341 fatcat:dxttdrqa5bdhxcpowfljlvcwle

An Experimental Comparison of SONC and SOS Certificates for Unconstrained Optimization [article]

Henning Seidler, Timo de Wolff
2018 arXiv   pre-print
As a main result, we carry out the first large-scale comparison 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.  ... 
arXiv:1808.08431v1 fatcat:rpq4gzqmg5h4vkd7qfc3opzmda

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

Hamza Fawzi, James Saunderson, Pablo A. Parrilo
2016 Mathematical programming  
First, 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.  ... 
doi:10.1007/s10107-015-0977-z fatcat:c3kfgr6yarhdhdyjlnh3pgyylu

A Semidefinite Hierarchy for Disjointly Constrained Multilinear Programming [article]

Kai Kellner
2016 arXiv   pre-print
Under an additional geometric condition, the NP-complete 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] .  ... 
arXiv:1603.03634v1 fatcat:happcknkmzacfbb4q2c774zxky

Pre- and Post-Processing Sum-of-Squares Programs in Practice

J. Lofberg
2009 IEEE Transactions on Automatic Control  
This paper describes how the 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(  ... 
doi:10.1109/tac.2009.2017144 fatcat:yufyuutivrh5rnulblzaqiklpu

Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

Hamza Fawzi, James Saunderson, Pablo A. Parrilo
2015 SIAM Journal on Optimization  
Our main result is a structure theorem where we show that any equivariant psd lift 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.  ... 
doi:10.1137/140966265 fatcat:vrjpdsih7bhonjslfwpjbwpe7a

f-vectors of 3-polytopes symmetric under rotations and rotary reflections [article]

Maren H. Ring, Robert Schüler
2020 arXiv   pre-print
We give a full answer 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.  ... 
arXiv:2002.00355v2 fatcat:4pzlsbd4xfbqzo7kyfaayuxo6i

Finding and Optimizing Certified, Collision-Free Regions in Configuration Space for Robot Manipulators [article]

Alexandre Amice, Hongkai Dai, Peter Werner, Annan Zhang, Russ Tedrake
2022 arXiv   pre-print
Our regions are generated by alternating between two convex optimization problems: (1) a simultaneous search 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  ... 
arXiv:2205.03690v1 fatcat:4afsyauzpjhn5denq7igvnmjla

Equivariant Semidefinite Lifts of Regular Polygons

Hamza Fawzi, James Saunderson, Pablo A. Parrilo
2017 Mathematics of Operations Research  
Our construction relies on finding a sparse 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.  ... 
doi:10.1287/moor.2016.0813 fatcat:rlzpjx525bb6jeeqhnaydc5dyq

Gram Spectrahedra [article]

Lynn Chua, Daniel Plaumann, Rainer Sinn, Cynthia Vinzant
2018 arXiv   pre-print
Representations 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.  ... 
arXiv:1608.00234v3 fatcat:vhz3ystdszg7pkx74mxo5nyxz4

General non-realizability certificates for spheres with linear programming [article]

Joao Gouveia, Antonio Macchia, Amy Wiebe
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.  ... 
arXiv:2109.15247v1 fatcat:6l4wch3v7bcnrdhmkc3kjcyvtu

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

Sadik Iliman, Timo de Wolff
2015 arXiv   pre-print
Additionally, these statements yield a completely new class 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.  ... 
arXiv:1402.0462v3 fatcat:7qkxa735vngpdfiac2cw7z7s7u

Amoebas, nonnegative polynomials and sums of squares supported on circuits

Sadik Iliman, Timo de Wolff
2016 Research in the Mathematical Sciences  
Additionally, these statements yield a completely new class 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.  ... 
doi:10.1186/s40687-016-0052-2 fatcat:a33nmfioqvb75jnqfuhou5pmca

Lower Bounds for Polynomials with Simplex Newton Polytopes Based on Geometric Programming [article]

Sadik Iliman, Timo de Wolff
2016 arXiv   pre-print
We provide new sufficient conditions 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.  ... 
arXiv:1402.6185v4 fatcat:nzp7v3qeivfcxjug4unp3ut5ui
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