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162
SUM OF SQUARES CERTIFICATES FOR CONTAINMENT OF H-POLYTOPES IN V-POLYTOPES
"... Abstract. Given an H-polytope P and a V-polytope Q, the decision problem whether P is contained in Q is co-NP-complete. This hardness remains if P is restricted to be a standard cube and Q is restricted to be the affine image of a cross polytope. While this hardness classification by Freund and Orli ..."
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Cited by 1 (0 self)
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and Orlin dates back to 1985, for general dimension there seems to be only limited progress on that problem so far. 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. These certificates can be computed
A NOTE ON THE NONEXISTENCE OF SUM OF SQUARES CERTIFICATES FOR THE BMV CONJECTURE
, 2010
"... The algebraic reformulation of the BMV conjecture is equivalent to a family of dimension-free tracial inequalities involving positive semidefinite matrices. Sufficient conditions for these to hold in the form of algebraic identities involving polynomials in noncom-muting variables have been given ..."
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by Markus Schweighofer and the second author. Later the existence of these certificates has been settled for all but one case, which is resolved in this note.
Sums of squares, moment matrices and optimization over polynomials
, 2008
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory ..."
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Cited by 154 (9 self)
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theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums
CERTIFICATE OF APPROVAL
, 1984
"... Application of least square cubic splines to the analysis of edges ..."
An exact duality theory for semidefinite programming based on sums of squares
, 2012
"... Farkas’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certifi ..."
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Cited by 13 (2 self)
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radical and sums of squares certificates from real algebraic geometry.
Global optimization of polynomials using gradient tentacles and sums of squares
- SIAM Journal on Optimization
"... We consider the problem of computing the global infimum of a real polynomial f on R n. Every global minimizer of f lies on its gradient variety, i.e., the algebraic subset of R n where the gradient of f vanishes. If f attains a minimum on R n, it is therefore equivalent to look for the greatest low ..."
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Cited by 26 (0 self)
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lower bound of f on its gradient variety. Nie, Demmel and Sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds. Based on these certificates, they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite
Quadratic-Time Certificates in Linear Algebra
, 2011
"... We present certificates for the positive semidefiniteness of an n × n matrix A, whose entries are integers of binary length log ‖A‖, that can be verified in O(n 2+ǫ (log ‖A‖) 1+ǫ) binary operations for any ǫ> 0. The question arises in Hilbert/Artin-based rational sum-of-squares certificates, i.e. ..."
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Cited by 4 (2 self)
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We present certificates for the positive semidefiniteness of an n × n matrix A, whose entries are integers of binary length log ‖A‖, that can be verified in O(n 2+ǫ (log ‖A‖) 1+ǫ) binary operations for any ǫ> 0. The question arises in Hilbert/Artin-based rational sum-of-squares certificates, i
CERTIFICATE OF APPROVAL
, 2014
"... Treatment plan optimization for rotating-shield brachytherapy ..."
Verifying nonlinear real formulas via sums of squares
- Theorem Proving in Higher Order Logics, TPHOLs 2007, volume 4732 of Lect. Notes in Comp. Sci
, 2007
"... Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates ..."
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Cited by 31 (3 self)
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Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares
Quantum Certificate Complexity
- In Proc. IEEE Conference on Computational Complexity
, 2003
"... Given a Boolean function f, we study two natural gener- alizations of the certificate complexity C (f): the randomized certificate complexity RC (f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC (f) as the square root of RC(f). We the ..."
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Cited by 13 (4 self)
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Given a Boolean function f, we study two natural gener- alizations of the certificate complexity C (f): the randomized certificate complexity RC (f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC (f) as the square root of RC(f). We
Results 1 - 10
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