The Search for the Maximum of a Polynomial

Uteshev, Alexei Yu.; Cherkasov, T. M.

doi:10.1006/jsco.1997.0190

Cited by 17 publications

(6 citation statements)

References 15 publications

(1 reference statement)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One may establish the relationship similar to (3.10) from the univariate case: between the matrix B given by (4.7) and the block Hankel matrices C and D from the §4 of the paper (Uteshev and Cherkasov, 1998). This allows one to use the minors of the matrix B constructed for g(x, y) = (x − t)(y − u)J (x, y) for the zero localization problem in R 2 (Bikker and Uteshev, 1999).…”

Section: Solution Of Example 54mentioning

confidence: 96%

See 1 more Smart Citation

On the Bézout Construction of the Resultant

Bikker¹,

Uteshev

1999

Journal of Symbolic Computation

View full text Add to dashboard Cite

This paper deals with some ideas of Bézout and his successors Poisson, Netto and Laurent for solving polynomial systems. We treat them from the determinantal and from the Gröbner basis point of view. This results in effective algorithms for constructing the multivariate resultant. Other problems of Elimination Theory are discussed: how to find an eliminant for a polynomial system, how to represent its zeroes as the rational functions of the roots of this eliminant and how to separate zeroes.

show abstract

Section: Solution Of Example 54mentioning

confidence: 96%

“…. , λ n of f (x) (Uteshev and Shulyak, 1992;Uteshev and Cherkasov, 1998). Construct the following Laurent expan-…”

Section: The Univariate Casementioning

confidence: 99%

On the Bézout Construction of the Resultant

Bikker¹,

Uteshev

1999

Journal of Symbolic Computation

View full text Add to dashboard Cite

show abstract

“…Since (24) and (25) are four at most fourth degree univariate polynomial minimization problems on the interval [−1, 1], it is easy to find their global minimum Since n = 2, we use Algorithm 3 to compute the global minimizer of X (α 1 , α 2 ) in eight iterations.…”

Section: If the Newton Direction D Does Not Exist Or Ifmentioning

confidence: 99%

“…The multivariate polynomial optimization problem has attracted some attention recently [6,12,14,16,17,18,25]. It has applications in signal processing [2,5,18,22,23]; merit functions of polynomial equations [6]; 0 − 1 integer, linear, and quadratic programs [12]; nonconvex quadratic programs [12]; and bilinear matrix inequalities [12].…”

Section: Introductionmentioning

confidence: 99%

Global Minimization of Normal Quartic Polynomials Based on Global Descent Directions

Wan²,

Yang³

2004

SIAM J. Optim.

View full text Add to dashboard Cite

Abstract.A normal quartic polynomial is a quartic polynomial whose fourth degree term coefficient tensor is positive definite. Its minimization problem is one of the simplest cases of nonconvex global optimization, and has engineering applications. We call a direction a global descent direction of a function at a point if there is another point with a lower function value along this direction. For a normal quartic polynomial, we present a criterion to find a global descent direction at a noncritical point, a saddle point, or a local maximizer. We give sufficient conditions to judge whether a local minimizer is global and give a method for finding a global descent direction at a local, but not global, minimizer. We also give a formula at a critical point and a method at a noncritical point to find a one-dimensional global minimizer along a global descent direction. Based upon these, we propose a global descent algorithm for finding a global minimizer of a normal quartic polynomial when n = 2. For the case n ≥ 3, we propose an algorithm for finding an -global minimizer. At each iteration of a second algorithm, a system of constrained nonlinear equations is solved. Numerical tests show that these two algorithms are promising.

show abstract

“…When the system formed by the first order conditions has infinitely many solutions, the critical points of the given polynomial may not be simply and clearly described. In Uteshev and Cherkasov (1998), some assumptions on the given polynomial are made so that the applicability is restricted. In Parrilo and Sturmfels (2003), the global minimization for a certain family of polynomials was investigated by using semidefinite programming.…”

mentioning

confidence: 99%

Global minimization of multivariate polynomials using nonstandard methods

Zeng

Xiao

2011

J Glob Optim

View full text Add to dashboard Cite

The purpose of this paper is to present two algorithms for global minimization of multivariate polynomials. For a multivariate real polynomial f , we provide an effective algorithm for deciding whether or not the infimum of f is finite. In the case of f having a finite infimum, the infimum of f can be accurately coded as (h; a, b), where h is a real polynomial in one variable, a and b is two rational numbers with a < b, and (h, a, b) stands for the only real root of h in the open interval ]a, b[. Moreover, another algorithm is provided to decide whether or not the infimum of f is attained when the infimum of f is finite. Our methods are called "nonstandard", because an infinitesimal element is introduced in our arguments.

show abstract

The Search for the Maximum of a Polynomial

Cited by 17 publications

References 15 publications

On the Bézout Construction of the Resultant

On the Bézout Construction of the Resultant

Global Minimization of Normal Quartic Polynomials Based on Global Descent Directions

Global minimization of multivariate polynomials using nonstandard methods

Contact Info

Product

Resources

About