Validating the Completeness of the Real Solution Set of a System of Polynomial Equations

Brake, Daniel A.; Hauenstein, Jonathan D.; Liddell, Alan C.

doi:10.1145/2930889.2930910

Cited by 11 publications

(10 citation statements)

References 65 publications

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“…We remark that, while x certifies t − c1 whenever x − y x < 1 2 , and each step of our proof is valid for all 0 < r < 1 2 , we can only have r 2 1−2r ≤ r whenever 0 < r < 1 3 . Therefore, using Lemma 3.2, we can guarantee that x + − y x+ ≤ x − y x when x − y x < 1 3 . Below, we need to further limit r to ensure that the bound update is an improvement.…”

Section: 4mentioning

confidence: 99%

“…Lemma 3.2. Let x + and y be defined as above, and assume that x − y x ≤ r for some r < 1 3 . Then x + − y x+ ≤ r 2 1−2r .…”

Section: 4mentioning

confidence: 99%

“…The optimal value of the semidefinite program is an algebraic number, but the study of the algebraic degree of the positive semidefinite cone [15] suggests that one cannot hope for easily computable and verifiable certificates from taking a purely symbolic computing approach to the semidefinite programming problems that come from sums-of-squares. Therefore, a number of authors have proposed hybrid methods that "round" or "project" efficiently computable but inexact numerical sum-of-squares certificates to rigorous rational ones [20,10,3]; see also [5,6,1].…”

mentioning

confidence: 99%

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Dual certificates and efficient rational sum-of-squares decompositions for polynomial optimization over compact sets

Davis¹,

David²

2021

Preprint

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We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual cone certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate certifies a full-dimensional convex cone of WSOS polynomials. As a result, rational WSOS certificates can be constructed from numerically computed dual certificates at little additional cost, without any rounding or projection steps applied to the numerical certificates. As an additional algorithmic application, we present an almost entirely numerical hybrid algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence.

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Section: 4mentioning

confidence: 99%

“…Lemma 3.2. Let x + and y be defined as above, and assume that x − y x ≤ r for some r < 1 3 . Then x + − y x+ ≤ r 2 1−2r .…”

Section: 4mentioning

confidence: 99%

mentioning

confidence: 99%

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Dual certificates and efficient rational sum-of-squares decompositions for polynomial optimization over compact sets

Davis¹,

David²

2021

Preprint

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“…In [25] a stopping criterion is presented to verify that a Pommaret basis has been computed from the kernels of moment matrices involved in Sum of Squares relaxation. In [10], a test based on sum-of-square decomposition is proposed to verify that polynomials vanishing on a subset of the semi-algebraic set are in the real radical.…”

Section: Introductionmentioning

confidence: 99%

Computing real radicals by moment optimization

Baldi,

Mourrain

2021

Preprint

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We present a new algorithm for computing the real radical of an ideal and, more generally, the -radical of , which is based on convex moment optimization. A truncated positive generic linear functional vanishing on the generators of is computed solving a Moment Optimization Problem (MOP). We show that, for a large enough degree of truncation, the annihilator of generates the real radical of . We give an effective, general stopping criterion on the degree to detect when the prime ideals lying over the annihilator are real and compute the real radical as the intersection of real prime ideals lying over .The method involves several ingredients, that exploit the properties of generic positive moment sequences. A new efficient algorithm is proposed to compute a graded basis of the annihilator of a truncated positive linear functional. We propose a new algorithm to check that an irreducible decomposition of an algebraic variety is real, using a generic real projection to reduce to the hypersurface case. There we apply the Sign Changing Criterion, effectively performed with an exact MOP. Finally we illustrate our approach in some examples.

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“…Subsequently, Ma et al (2016) generalized this algorithm to positive dimensional cases. Brake et al (2016) gave a method based on numerical algebraic geometry and sums of squares programming to certify that a set of polynomials generates the real radical. We emphasize that these algorithms compute real radicals in R[X] and hence return approximate encodings of those radicals.…”

Section: Introductionmentioning

confidence: 99%

On the Complexity of Computing Real Radicals of Polynomial Systems

Din

Yang

Zhi

2018

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

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Let f = (f 1 ,. .. , f s) be a sequence of polynomials in Q[X 1 ,. .. , X n ] of maximal degree D and V ⊂ C n be the algebraic set defined by f and r be its dimension. The real radical re f associated to f is the largest ideal which defines the real trace of V. When V is smooth, we show that re f , has a finite set of generators with degrees bounded by deg V. Moreover, we present a probabilistic algorithm of complexity (snD n) O(1) to compute the minimal primes of re f. When V is not smooth, we give a probabilistic algorithm of complexity s O(1) (nD) O(nr2 r) to compute rational parametrizations for all irreducible components of the real algebraic set V ∩ R n. Experiments are given to show the efficiency of our approaches.

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Validating the Completeness of the Real Solution Set of a System of Polynomial Equations

Cited by 11 publications

References 65 publications

Dual certificates and efficient rational sum-of-squares decompositions for polynomial optimization over compact sets

Dual certificates and efficient rational sum-of-squares decompositions for polynomial optimization over compact sets

Computing real radicals by moment optimization

On the Complexity of Computing Real Radicals of Polynomial Systems

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