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

Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli
2016 arXiv   pre-print
For binary polynomial optimization problems of degree 2d and an odd number of variables n, we prove that n+2d-1/2 levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value.  ...  We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy.  ...  The authors would like to express their gratitude to Alessio Benavoli for helpful discussions.  ... 
arXiv:1605.03019v1 fatcat:fa4fy753kndgvlk6pals7jjspq

Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems *

Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli
unpublished
For binary polynomial optimization problems of degree 2d and an odd number of variables n, we prove that (n + 2d − 1)/2 levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value  ...  We give two results concerning the power of the Sum-Of-Squares(SoS)/Lasserre hierarchy.  ...  The authors would like to express their gratitude to Alessio Benavoli for helpful discussions.  ... 
fatcat:xhfszlbijvdkfa6uzswzgsgf5y

Minimum total-squared-correlation quaternary signature sets: new bounds and optimal designs

Ming Li, Stella Batalama, Dimitris Pados, John Matyjas
2009 IEEE Transactions on Communications  
Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to  ...  Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching data sources, gathering and maintaining  ...  The lower bound is tight if there exists a quaternary Hadamard matrix of size + 1 while the upper bound is tight if ≤ − 1 and there exists a quaternary Hadamard matrix of size − 1. ■ We recall that for  ... 
doi:10.1109/tcomm.2009.12.080589 fatcat:l7owybnjojczpcwklaips3jity

Certification of Real Inequalities -- Templates and Sums of Squares [article]

Xavier Allamigeon, Stéphane Gaubert, Victor Magron, Benjamin Werner
2014 arXiv   pre-print
We consider the problem of certifying lower bounds for real-valued multivariate transcendental functions.  ...  Our general framework is to use different approximation methods to relax the original problem into polynomial optimization problems, which we solve by sparse sums of squares relaxations.  ...  We also define the cone Σd [x] of sums of squares of degree at most 2d. 2.1 Constrained Polynomial Optimization Problems and SOS We consider the general constrained polynomial optimization problem (POP  ... 
arXiv:1403.5899v2 fatcat:df2jhxchzrbyjhht6a4atd6gqu

Page 2367 of Mathematical Reviews Vol. , Issue 95d [page]

1995 Mathematical Reviews  
If the lower bound technique could be extended to bounded degree algebraic decision trees, a tight Q(nlogn) lower bound for this latter problem would be obtained.”  ...  2367 95d:68065 68Q25 68Q05 68Q20 Ramanan, Prakash (1-WCHS-C; Wichita, KS) A new lower bound technique and its application: tight lower bound for a polygon triangulation problem.  ... 

Certification of Bounds of Non-linear Functions: The Templates Method [chapter]

Xavier Allamigeon, Stéphane Gaubert, Victor Magron, Benjamin Werner
2013 Lecture Notes in Computer Science  
This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation.  ...  The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions.  ...  Acknowledgements The authors thank the anonymous referees for helpful comments and suggestions to improve this paper.  ... 
doi:10.1007/978-3-642-39320-4_4 fatcat:ovizhweikbhvnjzopvsbsd2zmm

Bounded Global Optimization for Polynomial Programming using Binary Reformulation and Linearization [article]

Joseph W. Norman
2012 arXiv   pre-print
This paper describes an approximate method for global optimization of polynomial programming problems with bounded variables.  ...  The method uses a reformulation and linearization technique to transform the original polynomial optimization problem into a pair of mixed binary-linear programs.  ...  This predictability of error bounds means that the method can generate hard interval bounds on the global solution to each polynomial optimization problem, conditioned on the feasibility of that problem  ... 
arXiv:1205.6459v1 fatcat:m3aetsbhzfhxzpouh27yfwvmpi

Sums of Squares on the Hypercube [article]

Grigoriy Blekherman, João Gouveia, James Pfeiffer
2014 arXiv   pre-print
Using the lower bounds for C we construct a family of globally nonnegative quartic polynomials, which are not sums of squares of rational functions of small degree.  ...  To our knowledge this is the first construction for Hilbert's 17th problem of a family of polynomials of bounded degree which need increasing degrees in rational function representations as the number  ...  We leverage the tightness of the bound on C into a construction of a globally nonnegative polynomial p of degree 4 in n variables such that ph is not a sum of squares for all sums of squares h of degree  ... 
arXiv:1402.4199v1 fatcat:epyglo5d6bbh7iqkfkh62rp4mu

Provably Safe Tolerance Estimation for Robot Arms via Sum-of-Squares Programming [article]

Weiye Zhao, Suqin He, Changliu Liu
2021 arXiv   pre-print
It is theoretically proved that the algorithm provides a tight lower bound of the joint tolerance.  ...  This paper presented an efficient algorithm to estimate the joint tolerance using sum-of-squares programming.  ...  The proposed method is proved to provide a tight lower bound of the joint tolerance.  ... 
arXiv:2104.08896v1 fatcat:3mcnbepb4jbprcbvdo4hfojdvq

Verifying Individual Fairness in Machine Learning Models [article]

Philips George John, Deepak Vijaykeerthy, Diptikalyan Saha
2020 arXiv   pre-print
Our objective is to construct verifiers for proving individual fairness of a given model, and we do so by considering appropriate relaxations of the problem.  ...  We construct verifiers which are sound but not complete for linear classifiers, and kernelized polynomial/radial basis function classifiers.  ...  Acknowledgements The authors would like to thank Dinesh Garg and Rishi Saket for helpful discussions.  ... 
arXiv:2006.11737v1 fatcat:xwo6dyuh3ng7hpivtlq6d4s66m

Certification of real inequalities: templates and sums of squares

Victor Magron, Xavier Allamigeon, Stéphane Gaubert, Benjamin Werner
2014 Mathematical programming  
We consider the problem of certifying lower bounds for real-valued multivariate transcendental functions.  ...  Our general framework is to use different approximation methods to relax the original problem into polynomial optimization problems, which we solve by sparse sums of squares relaxations.  ...  The resulting constrained polynomial optimization problems are solved with sums of squares relaxation from Lasserre hierarchy, by calling a semidefinite solver.  ... 
doi:10.1007/s10107-014-0834-5 fatcat:envrtolapvafxoltclmof32ssu

Sum-of-squares hierarchies for binary polynomial optimization [article]

Lucas Slot, Monique Laurent
2022 arXiv   pre-print
We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube 𝔹^n={0,1}^n.  ...  This hierarchy provides for each integer r ∈ℕ a lower bound f_(r) on the minimum f_min of f, given by the largest scalar λ for which the polynomial f - λ is a sum-of-squares on 𝔹^n with degree at most  ...  We wish to thank Sven Polak and Pepijn Roos Hoefgeest for several useful discussions. We also thank the anonymous referees for their helpful comments and suggestions.  ... 
arXiv:2011.04027v3 fatcat:io3h2ofz2bdktkh4ct3ma7ufqe

Bounds for the sum of distances of spherical sets of small size [article]

Alexander Barg, Peter Boyvalenkov, Maya Stoyanova
2021 arXiv   pre-print
We derive upper and lower bounds on the sum of distances of a spherical code of size N in n dimensions when N∼ n^α, 0<α≤ 2.  ...  We discuss asymptotic behavior of our bounds along with several examples of codes whose sum of distances closely follows the upper bound.  ...  Thus, two-distance tight frames form spherical codes of size N in R n that have asymptotically maximum sum of distances while minimizing the sum of squares of the inner products. 3.3.  ... 
arXiv:2105.03511v1 fatcat:oxokwsotofeqlbfxbmqqawsfi4

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.  ...  increasingly tight lower bounds for the (QAP).  ... 
doi:10.17535/crorr.2014.0001 fatcat:fngmhkpzyza5jjhe4ssc7k5st4

Approximating Optimal Binary Decision Trees

Micah Adler, Brent Heeringa
2011 Algorithmica  
We conclude by showing that our upper bound also holds for the DT problem with weighted tests.  ...  We give a (ln n + 1)-approximation for the decision tree (DT) problem. An instance of DT is a set of m binary tests T = (T1, . . . , Tm) and a set of n items X = (X1, . . . , Xn).  ...  We thank the anonymous reviewers for their insightful and helpful comments.  ... 
doi:10.1007/s00453-011-9510-9 fatcat:nrewe4n5ebbong3bk3gertznfa
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