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

Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli
2016 arXiv   pre-print
We introduce a method 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.  ... 
arXiv:1407.1746v2 fatcat:wxu7jef6gvc2nk3iq4kh3pukhq

On the Power of Symmetric LP and SDP Relaxations

James R. Lee, Prasad Raghavendra, David Steurer, Ning Tan
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
This result gives the first 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  ... 
doi:10.1109/ccc.2014.10 dblp:conf/coco/LeeRST14 fatcat:rljvsqsehvfmvnnjoz6wrtidhq

Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

Hamza Fawzi, James Saunderson, Pablo A. Parrilo
2015 SIAM Journal on Optimization  
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.  ...  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.  ... 
doi:10.1137/140966265 fatcat:vrjpdsih7bhonjslfwpjbwpe7a

From combinatorial optimization to real algebraic geometry and back

Janez Povh
2014 Croatian Operational Research Review  
The latter 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.  ... 
doi:10.17535/crorr.2014.0001 fatcat:fngmhkpzyza5jjhe4ssc7k5st4

On the construction of converging hierarchies for polynomial optimization based on certificates of global positivity [article]

Amir Ali Ahmadi, Georgina Hall
2018 arXiv   pre-print
In recent years, techniques based on convex optimization and real algebra that produce converging 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  ... 
arXiv:1709.09307v2 fatcat:bnks6dbefjb7hmatquefb2dy2q

Tight Sum-of-Squares lower bounds for binary polynomial optimization problems [article]

Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli
2016 arXiv   pre-print
We give two results concerning the power 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.  ... 
arXiv:1605.03019v1 fatcat:fa4fy753kndgvlk6pals7jjspq

Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy [article]

Marek Tyburec, Jan Zeman, Martin Kružík, Didier Henrion
2020 arXiv   pre-print
This 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) .  ... 
arXiv:2009.12560v1 fatcat:fosmdfwht5eyratttcahd5mvsy

Query complexity in expectation [article]

Jedrzej Kaniewski, Troy Lee, Ronald de Wolf (CWI, University of Amsterdam)
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 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] .  ... 
arXiv:1411.7280v1 fatcat:qqpeu43kjnejfg5ipo24zxdkze

Geometry of 3D Environments and Sum of Squares Polynomials

Amir Ali Ahmadi, Georgina Hall, Ameesh Makadia, Vikas Sindhwani
2017 Robotics: Science and Systems XIII  
We use algebraic techniques from 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.  ... 
doi:10.15607/rss.2017.xiii.071 dblp:conf/rss/AhmadiHMS17 fatcat:qcefmbj4lveqdjqbnmlouijz54

Query Complexity in Expectation [chapter]

Jedrzej Kaniewski, Troy Lee, Ronald de Wolf
2015 Lecture Notes in Computer Science  
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  ...  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] .  ... 
doi:10.1007/978-3-662-47672-7_62 fatcat:a6fesbjv45fi3ehtxfmewvcj24

Geometry of 3D Environments and Sum of Squares Polynomials [article]

Amir Ali Ahmadi, Georgina Hall, Ameesh Makadia, Vikas Sindhwani
2017 arXiv   pre-print
We use algebraic techniques from 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.  ... 
arXiv:1611.07369v3 fatcat:vzscnfcylfafdf444vmxdpaspq

Semidefinite Programming and Nash Equilibria in Bimatrix Games [article]

Amir Ali Ahmadi, Jeffrey Zhang
2019 arXiv   pre-print
Finally, we show the connection between our SDP and the first level 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.  ... 
arXiv:1706.08550v3 fatcat:5zwr6lcukbhu3m3fvv54zba2ri

A numerical evaluation of the bounded degree sum-of-squares hierarchy of Lasserre, Toh, and Yang on the pooling problem

Ahmadreza Marandi, Joachim Dahl, Etienne de Klerk
2017 Annals of Operations Research  
The 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.  ... 
doi:10.1007/s10479-017-2407-5 fatcat:em25wdtc5jh5pjcurxafbrzj5u

The sum-of-squares hierarchy on the sphere and applications in quantum information theory

Kun Fang, Hamza Fawzi
2020 Mathematical programming  
We consider the problem 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 .  ... 
doi:10.1007/s10107-020-01537-7 fatcat:dvmmitjc7veifescq5t3pdbn6e

The matching problem has no small symmetric SDP [article]

Gábor Braun, Jonah Brown-Cohen, Arefin Huq, Sebastian Pokutta, Prasad Raghavendra, Aurko Roy, Benjamin Weitz, Daniel Zink
2016 arXiv   pre-print
We answer this question negatively 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).  ... 
arXiv:1504.00703v5 fatcat:vddxee5knjeehp3nlhgvzrt5ia
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