Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems

Möller, H. Michael; Stetter, Hans J.

doi:10.1007/s002110050122

Cited by 97 publications

(92 citation statements)

References 13 publications

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“…This problem has received considerable attention in the literature. We present the so-called eigenvalue method (also known as the Stetter-Möller method [92]) which relates the points of V C (I) to the eigenvalues of the multiplication operators in the quotient space R[x]/I. See, e.g., [19,37,144] for a detailed account on this method and various other methods for solving systems of polynomial equations.…”

Section: Sums Of Squares Moments and Polynomial Optimization 17mentioning

confidence: 99%

Sums of Squares, Moment Matrices and Optimization Over Polynomials

Laurent

2008

The IMA Volumes in Mathematics and Its Applications

631

721

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Abstract. 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 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 of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zero-dimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.

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Section: Sums Of Squares Moments and Polynomial Optimization 17mentioning

confidence: 99%

Sums of Squares, Moment Matrices and Optimization Over Polynomials

Laurent

2008

The IMA Volumes in Mathematics and Its Applications

631

721

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“…The matrix A x i is commonly called the ith companion matrix [21], the multiplication table [31] or the representation matrix [22] for the variable x i . The matrices A x i have the following natural properties.…”

Section: Algebraic Backgroundmentioning

confidence: 99%

“…The problem of finding a global minimum of a polynomial from this class can be reformulated as an eigenvalue problem by applying the Stetter-Möller matrix method [8,21]. This yields a set of real nonsymmetric large commuting matrices A p , A x 1 , .…”

Section: Introductionmentioning

confidence: 99%

Polynomial optimization and a Jacobi–Davidson type method for commuting matrices

Bleylevens

Hochstenbach

Peeters

2013

Applied Mathematics and Computation

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In this paper we introduce an new Jacobi-Davidson type eigenvalue solver for a set of commuting matrices, called JD-COMM, used for the global optimization of so-called Minkowski-norm dominated polynomials in several variables. The Stetter-Möller matrix method yields such a set of real non-symmetric commuting matrices since it reformulates the optimization problem as an eigenvalue problem. A drawback of this approach is that the matrix most relevant for computing the global optimum of the polynomial under investigation is usually large and only moderately sparse. However, the other matrices are generally much sparser and have the same eigenvectors because of the commutativity. This fact is used to design the JDCOMM method for this problem: the most relevant matrix is used only in the outer loop and the sparser matrices are exploited in the solution of the correction equation in the inner loop to greatly improve the efficiency of the method. Some numerical examples demonstrate that the method proposed in this paper is more efficient than approaches that work on the main matrix (standard Jacobi-Davidson and implicitly restarted Arnoldi), as well as conventional solvers for computing the global optimum, i.e., SOSTOOLS, GloptiPoly, and PHCpack.

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“…[24]), where the problem of solving nonlinear equations is reduced to an eigenvalue problem (see also [21]). Here, H-bases can be used to overcome representation singularities, which are known from [25] to cause severe numerical problems, as is shown by the following modification of the above example.…”

Section: As Möller Pointed Out In [18]mentioning

confidence: 99%

Gröbner bases, H–bases and interpolation

Sauer

2000

Trans. Amer. Math. Soc.

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Abstract. The paper is concerned with a construction for H-bases of polynomial ideals without relying on term orders. The main ingredient is a homogeneous reduction algorithm which orthogonalizes leading terms instead of completely canceling them. This allows for an extension of Buchberger's algorithm to construct these H-bases algorithmically. In addition, the close connection of this approach to minimal degree interpolation, and in particular to the least interpolation scheme due to de Boor and Ron, is pointed out.

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Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems

Cited by 97 publications

References 13 publications

Sums of Squares, Moment Matrices and Optimization Over Polynomials

Sums of Squares, Moment Matrices and Optimization Over Polynomials

Polynomial optimization and a Jacobi–Davidson type method for commuting matrices

Gröbner bases, H–bases and interpolation

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