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Sum-of-squares hierarchy lower bounds for symmetric formulations
[article]

2016
*
arXiv
*
pre-print

We introduce a method

arXiv:1407.1746v2
fatcat:wxu7jef6gvc2nk3iq4kh3pukhq
*for*proving*Sum*-*of*-*Squares*(SoS)/ Lasserre*hierarchy**lower**bounds*when the initial problem*formulation*exhibits a high degree*of*symmetry. ... More precisely, we give a short elementary proof*of*Grigoriev/Laurent*lower**bound**for*finding the integer cut polytope*of*the complete graph. ... The authors would like to express their gratitude to Ola Svensson*for*helpful discussions and ideas regarding this paper. ...##
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On the Power of Symmetric LP and SDP Relaxations

2014
*
2014 IEEE 29th Conference on Computational Complexity (CCC)
*

This result gives the first

doi:10.1109/ccc.2014.10
dblp:conf/coco/LeeRST14
fatcat:rljvsqsehvfmvnnjoz6wrtidhq
*lower**bounds**for**symmetric*SDP relaxations*of*Max CSPs, and indicates that the*sum*-*of*-*squares*method provides the "right" SDP relaxation*for*this class*of*problems. ... Concretely,*for*k < n/4, we show that k-rounds*of**sum*-ofsquares / Lasserre relaxations*of*size k n k achieve best-possible approximation guarantees*for*Max CSPs among all*symmetric*SDP relaxations*of*size ... Combined with known*lower**bounds**for**sum*-*of*-*squares*relaxations [17] - [19] , this result implies the first explicit*lower**bounds**for*general*symmetric*SDP relaxations*of*natural optimization problems ...##
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Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

2015
*
SIAM Journal on Optimization
*

We use this framework to study two well-known families

doi:10.1137/140966265
fatcat:vrjpdsih7bhonjslfwpjbwpe7a
*of*polytopes, namely the parity polytope and the cut polytope, and we prove exponential*lower**bounds**for*equivariant psd lifts*of*these polytopes. ... 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 ... The authors would like to thank the anonymous referees*for*their thorough comments which helped improve the paper, in particular Theorem 1.6. ...##
###
From combinatorial optimization to real algebraic geometry and back

2014
*
Croatian Operational Research Review
*

The latter

doi:10.17535/crorr.2014.0001
fatcat:fngmhkpzyza5jjhe4ssc7k5st4
*formulation*enables a*hierarchy**of*approximations which rely on results from polynomial optimization, a sub-field*of*real algebraic geometry. ... We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone*of*completely positive matrices. ... We can use a result, first*formulated*in [39] , that characterizes the monomials that can appear in a*sum**of**squares*representation. ...##
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On the construction of converging hierarchies for polynomial optimization based on certificates of global positivity
[article]

2018
*
arXiv
*
pre-print

In recent years, techniques based on convex optimization and real algebra that produce converging

arXiv:1709.09307v2
fatcat:bnks6dbefjb7hmatquefb2dy2q
*hierarchies**of**lower**bounds**for*polynomial minimization problems have gained much popularity. ... More precisely, we show that any inner approximation to the cone*of*positive homogeneous polynomials that is arbitrarily tight can be turned into a converging*hierarchy**of**lower**bounds**for*general polynomial ... We are grateful to Pablo Parrilo*for*very insightful comments, particularly as regards Section 4 and the observation that any form can be made even by only doubling the number*of*variables and the degree ...##
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Tight Sum-of-Squares lower bounds for binary polynomial optimization problems
[article]

2016
*
arXiv
*
pre-print

We give two results concerning the power

arXiv:1605.03019v1
fatcat:fa4fy753kndgvlk6pals7jjspq
*of*the*Sum*-*of*-*Squares*(SoS)/Lasserre*hierarchy*. ... We disprove this conjecture and derive*lower*and upper*bounds**for*the rank. ... The authors would like to express their gratitude to Alessio Benavoli*for*helpful discussions. ...##
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Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy
[article]

2020
*
arXiv
*
pre-print

This

arXiv:2009.12560v1
fatcat:fosmdfwht5eyratttcahd5mvsy
*formulation*is subsequently solved using the Lasserre moment-*sum*-*of*-*squares**hierarchy*, generating a sequence*of*outer convex approximations that monotonically converges from below to the optimum*of*... These theoretical findings are illustrated on several examples*of*topology optimization*of*frames and shells,*for*which we observe that the*hierarchy*converges in a finite (rather small) number*of*steps ... Acknowledgements We thank Edita Dvořáková*for*providing us with her implementation*of*the MITC4 shell elements (Dvořáková, 2015) . ...##
###
Query complexity in expectation
[article]

2014
*
arXiv
*
pre-print

We exactly characterize both the randomized and the quantum query complexity by two polynomial degrees, the nonnegative literal degree and the

arXiv:1411.7280v1
fatcat:qqpeu43kjnejfg5ipo24zxdkze
*sum*-*of*-*squares*degree, respectively. ... Since query complexity can be used to upper*bound*communication complexity*of*related functions, we can derive some upper*bounds*on psd extension complexity by constructing efficient quantum query algorithms ... us a version*of*[LRS14] . ...##
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Geometry of 3D Environments and Sum of Squares Polynomials

2017
*
Robotics: Science and Systems XIII
*

We use algebraic techniques from

doi:10.15607/rss.2017.xiii.071
dblp:conf/rss/AhmadiHMS17
fatcat:qcefmbj4lveqdjqbnmlouijz54
*sum**of**squares*optimization that reduce all these tasks to semidefinite programs*of*small size and present numerical experiments in realistic scenarios. ... Fig. 1 : Sublevel sets*of*sos-convex polynomials*of*increasing degree (left); sublevel sets*of*sos polynomials*of*increasing nonconvexity (middle); growth and shrinkage*of*an sos-body with sublevel sets ... ACKNOWLEDGEMENTS We thank Erwin Coumans, Mrinal Kalakrishnan and Vincent Vanhoucke*for*several technically insightful discussions and guidance. ...##
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Query Complexity in Expectation
[chapter]

2015
*
Lecture Notes in Computer Science
*

Since query complexity can be used to upper

doi:10.1007/978-3-662-47672-7_62
fatcat:a6fesbjv45fi3ehtxfmewvcj24
*bound*communication complexity*of*related functions, we can derive some upper*bounds*on psd extension complexity by constructing efficient quantum query algorithms ... We observe that the quantum complexity can be unboundedly smaller than the classical complexity*for*some functions, but can be at most polynomially smaller*for*functions with range {0, 1}. ... us a version*of*[LRS14] . ...##
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Geometry of 3D Environments and Sum of Squares Polynomials
[article]

2017
*
arXiv
*
pre-print

We use algebraic techniques from

arXiv:1611.07369v3
fatcat:vzscnfcylfafdf444vmxdpaspq
*sum**of**squares*optimization that reduce all these tasks to semidefinite programs*of*small size and present numerical experiments in realistic scenarios. ...*of*two convex basic semalgebraic sets that overlap, and tight containment*of*the union*of*several basic semialgebraic sets with a single convex one. ... ACKNOWLEDGEMENTS We thank Erwin Coumans, Mrinal Kalakrishnan and Vincent Vanhoucke*for*several technically insightful discussions and guidance. ...##
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Semidefinite Programming and Nash Equilibria in Bimatrix Games
[article]

2019
*
arXiv
*
pre-print

Finally, we show the connection between our SDP and the first level

arXiv:1706.08550v3
fatcat:5zwr6lcukbhu3m3fvv54zba2ri
*of*the Lasserre/*sum**of**squares**hierarchy*. ... We introduce an SDP relaxation*for*a quadratic programming*formulation**of*the Nash equilibrium (NE) problem and provide a number*of*valid inequalities to improve the quality*of*the relaxation. ... We would like to thank Ilan Adler, Costis Daskalakis, Georgina Hall, Ramon van Handel, and Robert Vanderbei*for*insightful exchanges. ...##
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A numerical evaluation of the bounded degree sum-of-squares hierarchy of Lasserre, Toh, and Yang on the pooling problem

2017
*
Annals of Operations Research
*

The

doi:10.1007/s10479-017-2407-5
fatcat:em25wdtc5jh5pjcurxafbrzj5u
*bounded*degree*sum*-*of*-*squares*(BSOS)*hierarchy**of*Lasserre et al. ... (EURO J Comput Optim 1-31, 2015) constructs*lower**bounds**for*a general polynomial optimization problem with compact feasible set, by solving a sequence*of*semi-definite programming (SDP) problems. ... Acknowledgements The authors would like to thank Claudia D'Ambrosio and Ruth Misener*for*useful discussions and providing us some references. ...##
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The sum-of-squares hierarchy on the sphere and applications in quantum information theory

2020
*
Mathematical programming
*

We consider the problem

doi:10.1007/s10107-020-01537-7
fatcat:dvmmitjc7veifescq5t3pdbn6e
*of*maximizing a homogeneous polynomial on the unit sphere and its*hierarchy**of**sum*-*of*-*squares*relaxations. ... By exploiting the duality relation between*sums**of**squares*and the Doherty-Parrilo-Spedalieri*hierarchy*in quantum information theory, we show that our result generalizes to nonquadratic polynomials the ... We will actually prove a more general result giving*bounds*on the performance*of*the*sum*-*of*-*squares**hierarchy**for*all values*of*the level . ...##
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The matching problem has no small symmetric SDP
[article]

2016
*
arXiv
*
pre-print

We answer this question negatively

arXiv:1504.00703v5
fatcat:vddxee5knjeehp3nlhgvzrt5ia
*for**symmetric*SDPs: any*symmetric*SDP*for*the matching problem has exponential size. ... We also show that an O(k)-round Lasserre SDP relaxation*for*the metric traveling salesperson problem yields at least as good an approximation as any*symmetric*SDP relaxation*of*size n^k. ... We now turn a G-coordinate-*symmetric*SDP*formulation*into a*symmetric**sum**of**squares*representation over a small set*of*basis functions. Lemma 2.3 (*Sum**of**squares**for*a*symmetric*SDP*formulation*). ...
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