Separation of real algebraic sets and the Łojasiewicz exponent

Kurdyka, Krzysztof; Spodzieja, Stanisław

doi:10.1090/s0002-9939-2014-12061-2

Cited by 30 publications

(30 citation statements)

References 32 publications

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“…In case K = R, a formula for computing L (f ) was given by Kuo in [12]. Furthermore, we have the following result (see also [4,6,10,11,15,21]). In particular, we have L (f ) ≤ (d − 1) 2 + 1.…”

Section: Effective Estimates For Lojasiewicz Exponentsmentioning

confidence: 77%

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Topological invariants of plane curve singularities: Polar quotients and Łojasiewicz gradient exponents

2019

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In this paper, we study polar quotients and Lojasiewicz exponents of plane curve singularities, which are not necessarily reduced. We first show that the polar quotients is a topological invariant. We next prove that the Lojasiewicz gradient exponent can be computed in terms of the polar quotients, and so it is also a topological invariant. As an application, we give effective estimates of the Lojasiewicz exponents in the gradient and classical inequalities of polynomials in two (real or complex) variables.

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Section: Effective Estimates For Lojasiewicz Exponentsmentioning

confidence: 77%

“…M, AND PHI-DŨNG HOÀNG Proof. The first inequality is an immediate consequence of the proof of Theorem 2.2 in [21] (see also [15]). The second one can be deduced from Theorem 4.9.…”

Section: Effective Estimates For Lojasiewicz Exponentsmentioning

confidence: 79%

Topological invariants of plane curve singularities: Polar quotients and Łojasiewicz gradient exponents

2019

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“…We recall the following Hölder-type error bound with an explicit exponent (see [31][32][33]). Let R be a positive number such that S contains an element x with x ≤ R. Then, there exists a constant c > 0 such that…”

Section: Theorem 24mentioning

confidence: 99%

Stability and Genericity for Semi-algebraic Compact Programs

Lee

Phạm

2016

J Optim Theory Appl

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In this paper, we consider the class of polynomial optimization problems over semi-algebraic compact sets, in which the objective functions are perturbed, while the constraint functions are kept fixed. Under certain assumptions, we establish some stability properties of the global solution map, of the Karush-Kuhn-Tucker set-valued map, and of the optimal value function for all problems in the class. It is shown that, for almost every problem in the class, there is a unique optimal solution for which the global quadratic growth condition and the strong second-order sufficient conditions hold. Furthermore, under local perturbations to the objective function, the optimal solution and the optimal value function (respectively, the Karush-Kuhn-Tucker setvalued map) vary smoothly (respectively, continuously) and the set of active constraint indices is constant. As a nice consequence, for almost every polynomial optimization problem, there is a unique optimal solution, which can be approximated arbitrarily closely by solving a sequence of semi-definite programs.Keywords Semi-algebraic program · Stability · Genericity · Polynomial · Semi-definite program Mathematics Subject Classification 90C26 · 90C31 · 49J45 · 49J50 · 49K40 B Gue Myung Lee

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“…Since the sphere S p−1 is a compact set, it follows from the Lojasiewicz inequality (see, for example, [23,34]) that there is a constant c > 0 such that…”

Section: Nonsmooth Lojasiewicz Gradient Inequality For the Largest Eimentioning

confidence: 99%

Łojasiewicz-type inequalities with explicit exponents for the largest eigenvalue function of real symmetric polynomial matrices

Đinh

Phạm

2016

Int. J. Math.

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Let [Formula: see text] be a real symmetric polynomial matrix of order [Formula: see text] and let [Formula: see text] be the largest eigenvalue function of the matrix [Formula: see text] We denote by [Formula: see text] the Clarke subdifferential of [Formula: see text] at [Formula: see text] In this paper, we first give the following nonsmooth version of Łojasiewicz gradient inequality for the function [Formula: see text] with an explicit exponent: For any [Formula: see text] there exist [Formula: see text] and [Formula: see text] such that we have for all [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] is a function introduced by D’Acunto and Kurdyka: [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text] Then we establish some local and global versions of Łojasiewicz inequalities which bound the distance function to the set [Formula: see text] by some exponents of the function [Formula: see text].

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Separation of real algebraic sets and the Łojasiewicz exponent

Abstract: We discuss several aspects of Łojasiewicz inequalities, namely local and global versions, and the relations between the gradient inequality and regular separation of real algebraic sets. We give effective estimates for Łojasiewicz's exponents in the real and complex setting.

Cited by 30 publications

References 32 publications

Topological invariants of plane curve singularities: Polar quotients and Łojasiewicz gradient exponents

Topological invariants of plane curve singularities: Polar quotients and Łojasiewicz gradient exponents

Stability and Genericity for Semi-algebraic Compact Programs

Łojasiewicz-type inequalities with explicit exponents for the largest eigenvalue function of real symmetric polynomial matrices

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