Generalized normal forms and polynomial system solving

Mourrain, Bernard; Trébuchet, Philippe

doi:10.1145/1073884.1073920

Cited by 46 publications

(55 citation statements)

References 23 publications

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“…In [16], one strives to enumerate all the solutions of a KKT system, not only the global optimum, as all the KKT solutions will be meaningful in this application. Indeed, the authors develop special algorithms for that purpose: e.g., the subdivision methods proposed by Mourrain and Pavone [40], and the generalized normal forms algorithms designed by Mourrain and Trébuchet [41]. However, the shortcomings of these methods are apparent if the degree of the polynomial is high.…”

mentioning

confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

159

177

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Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we only accept a block update that achieves the maximum improvement, hence the name of our new search method: Maximum Block Improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proven. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem, thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.Key words. block coordinate descent, polynomial optimization problem, tensor form AMS subject classifications. 90C26, 90C30, 15A69, 49M271. Introduction. The optimization models whose objective and constraints are polynomial functions have recently attracted much research attention. This is in part due to an increased demand on the application side (cf. the sample applications in numerical linear algebra [50,26,28], material sciences [57], quantum physics [9,18], and signal processing [16,4,53]), and in part due to its own strong theoretical appeal. Indeed, polynomial optimization is a challenging task; at the same time it is rich enough to be fruitful. For instance, even the simplest instances of polynomial optimization, such as maximizing a cubic polynomial over a sphere, is NP-hard (Nesterov [42]). However, the problem is so elementary that it can be even attempted in an undergraduate calculus class. For readers interested in polynomial optimization with simple constraints, we refer to De Klerk [12] for a survey on the computational complexity of optimizing various classes of polynomial functions over a simplex, hypercube or sphere. In particular, De Klerk et al.[13] designed a polynomial-time approximation scheme (PTAS) for minimizing polynomials of fixed degree over the simplex.So far, a few results have been obtained for approximation algorithms with guaranteed worst-case performance ratios for higher degree polynomial optimization problems. Luo and Zhang [39] derived a polynomial-time approximation algorithm to optimize a multivariate quartic polynomial over a region defined by quadratic inequalities.

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mentioning

confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

159

177

View full text Add to dashboard Cite

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“…It turns out that, by adding the positive semidefiniteness constraint, the method of [18] can be adapted and extended for computing the (finite) real variety V R (I); this will be treated in detail in the follow-up paper [9]. Summarizing, our results provide a unified treatment of the computation of real and complex roots either by means of moment matrices or by means of a dual form characterization as in [9], [17] and [18].…”

Section: Motivation and Contributionmentioning

confidence: 98%

“…This motivates the second contribution of this paper, which is to relate the proposed method based on moment matrices to existing methods and, in particular, to the method of [18] (and [17]) for the (finite) complex variety. It turns out that, by adding the positive semidefiniteness constraint, the method of [18] can be adapted and extended for computing the (finite) real variety V R (I); this will be treated in detail in the follow-up paper [9].…”

Section: Motivation and Contributionmentioning

confidence: 99%

“…In particular, algorithms have been proposed for constructing border bases of I leading to general (stable by division) bases of R[x]/I (see [6,Chapter 4], [8] and [23]). Another normal form algorithm is proposed by Mourrain [14] (see also [15,17]) leading to more general (namely, connected to 1) bases of R[x]/I. The moment-matrix approach of this paper allows the computation of general polynomial bases of R[x]/I (or of R[x]/I(V R (I)) as explained in [10]).…”

Section: Multiplication Operators Given a Polynomial H ∈ R[x]mentioning

confidence: 99%

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A Unified Approach to Computing Real and Complex Zeros of Zero-Dimensional Ideals

Lasserre

Laurent

Rostalski

2008

The IMA Volumes in Mathematics and Its Applications

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Abstract. In this paper we propose a unified methodology for computing the setWe show how moment matrices, defined in terms of a given set of generators of the ideal I, can be used to (numerically) find not only the real variety V R (I), as shown in the authors' previous work, but also the complex variety V C (I), thus leading to a unified treatment of the algebraic and real algebraic problem. In contrast to the real algebraic version of the algorithm, the complex analogue only uses basic numerical linear algebra because it does not require positive semidefiniteness of the moment matrix and so avoids semidefinite programming techniques. The links between these algorithms and other numerical algebraic methods are outlined and their stopping criteria are related.

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“…The family F := {f m | m ∈ ∂B} is called a rewriting family for B in [30,32]. Using F , one can express all border monomials in ∂B as linear combinations of monomials in B modulo the ideal F .…”

Section: Border Bases and Normal Formsmentioning

confidence: 99%

The Approach of Moments for Polynomial Equations

Laurent

Rostalski

2011

International Series in Operations Research &Amp; Management Science

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Summary. In this chapter we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of polynomials and the dual theory of moments. A crucial ingredient is a semidefinite characterization of the real radical ideal, consisting of all polynomials with the same real zero set as the system of polynomials to be solved. Combining this characterization with ideas from commutative algebra, (numerical) linear algebra and semidefinite optimization yields a new class of real algebraic algorithms. This chapter sheds some light on the underlying theory and the link to polynomial optimization.

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Generalized normal forms and polynomial system solving

Cited by 46 publications

References 23 publications

Maximum Block Improvement and Polynomial Optimization

Maximum Block Improvement and Polynomial Optimization

A Unified Approach to Computing Real and Complex Zeros of Zero-Dimensional Ideals

The Approach of Moments for Polynomial Equations

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