Abstract Cones of Positive Polynomials and Their Sums of Squares Relaxations

2010 ◽  
pp. 41-50 ◽  
Author(s):  
Roland Hildebrand
Keyword(s):  
Sums Of Squares ◽  
Positive Polynomials

scholarly journals There Are Many More Positive Maps Than Completely Positive Maps

2017 ◽  
Vol 2019 (11) ◽  
pp. 3313-3375 ◽  
Author(s):  
Igor Klep ◽  
Scott McCullough ◽  
Klemen Šivic ◽  
Aljaž Zalar
Keyword(s):  
Algebraic Geometry ◽  
Main Tool ◽  
Positive Semidefinite ◽  
Sums Of Squares ◽  
Completely Positive Maps ◽  
Positive Semidefinite Matrices ◽  
Positive Polynomials ◽  
Completely Positive ◽  
Positive Maps ◽  
Matrix Spaces

Abstract A $\ast$-linear map $\Phi$ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations $I_n\otimes \Phi$ are positive. In this article, quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps that are not completely positive is given.

scholarly journals Minkowski’s inequality and sums of squares

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Péter Frenkel ◽  
Péter Horváth
Keyword(s):  
Sums Of Squares ◽  
Minkowski’S Inequality ◽  
Power Mean ◽  
Positive Polynomials

AbstractPositive polynomials arising from Muirhead’s inequality, from classical power mean and elementary symmetric mean inequalities and from Minkowski’s inequality can be rewritten as sums of squares.

scholarly journals Convexifying Positive Polynomials and Sums of Squares Approximation

2015 ◽  
Vol 25 (4) ◽  
pp. 2512-2536 ◽  
Author(s):  
Krzysztof Kurdyka ◽  
StanisŁaw Spodzieja
Keyword(s):  
Sums Of Squares ◽  
Positive Polynomials

Sums of Squares and Positive Polynomials

2021 ◽  
pp. 15-26
Author(s):  
Victoria Powers
Keyword(s):  
Sums Of Squares ◽  
Positive Polynomials

Approximating Positive Polynomials Using Sums of Squares

2003 ◽  
Vol 46 (3) ◽  
pp. 400-418 ◽  
Author(s):  
M. Marshall
Keyword(s):  
Moment Problem ◽  
Sums Of Squares ◽  
General Context ◽  
Positive Polynomials ◽  
Quadratic Module ◽  
Algebraic Sets ◽  
Compact Case ◽  
The Relationship

AbstractThe paper considers the relationship between positive polynomials, sums of squares and the multi-dimensional moment problem in the general context of basic closed semi-algebraic sets in real n-space. The emphasis is on the non-compact case and on quadratic module representations as opposed to quadratic preordering presentations. The paper clarifies the relationship between known results on the algebraic side and on the functional-analytic side and extends these results in a variety of ways.

scholarly journals Towards a computational proof of Vizing's conjecture using semidefinite programming and sums-of-squares

2021 ◽  
Vol 107 ◽  
pp. 67-105
Author(s):  
Elisabeth Gaar ◽  
Daniel Krenn ◽  
Susan Margulies ◽  
Angelika Wiegele
Keyword(s):  
Semidefinite Programming ◽  
Sums Of Squares

scholarly journals Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

2021 ◽  
Author(s):  
Mareike Dressler ◽  
Adam Kurpisz ◽  
Timo de Wolff
Keyword(s):  
Computer Science ◽  
Affine Transformation ◽  
Optimization Problems ◽  
Polynomial Optimization ◽  
Theoretical Computer Science ◽  
Sums Of Squares ◽  
Theoretical Computer ◽  
Complexity Bounds ◽  
Polynomial Optimization Problems ◽  
Transformation Of Variables

AbstractVarious key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most $$n+d$$ n + d . Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most $$n^{O(d)}$$ n O ( d ) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

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