We prove an upper bound on the degree complexity of Putinar's
Positivstellensatz. This bound is much worse than the one obtained previously
for Schm\"udgen's Positivstellensatz but it depends on the same parameters. As
a consequence, we get information about the convergence rate of Lasserre's
procedure for optimization of a polynomial subject to polynomial constraints

Nie, Jiawang

Schweighofer, Markus

English

arXiv

We prove an upper bound on the degree complexity of Putinar's Positivstellensatz. This bound is much worse than the one obtained previously for Schmüdgen's Positivstellensatz but it depends on the same parameters. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints.publishe

KOPS - The Institutional Repository of the University of Konstanz

On the complexity of Putinar's Positivstellensatz

AbstractLet S={x∈Rn∣g1(x)≥0,…,gm(x)≥0} be a basic closed semialgebraic set defined by real polynomials gi. Putinar's Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S possesses a representation f=∑i=0mσigi where g0≔1 and each σi is a sum of squares of polynomials. Such a representation is a certificate for the nonnegativity of f on S. We give a bound on the degrees of the terms σigi in this representation which depends on the description of S, the degree of f and a measure of how close f is to having a zero on S. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints

Elsevier - Publisher Connector 

On the complexity of Putinar's Positivstellensatz 

International audienceWe prove an upper bound on the degree complexity of Putinar's Positivstellensatz. This bound is much worse than the one obtained previously for Schmüdgen's Positivstellensatz but it depends on the same parameters. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints

HAL-Rennes 1

Let S = {x ∈ Rn | g1(x)  ≥ 0,..., gm(x)  ≥ 0} be a basic closed semialgebraic set defined by real polynomials gi. Putinar’s Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S posesses a representation f = Pm i=0 σigi where g0: = 1 and each σi is a sum of squares of polynomials. Such a representation is a certificate for the nonnegativity of f on S. We give a bound on the degrees of the terms σigi in this representation which depends on the description of S, the degree of f and a measure of how close f is to having a zero on S. As a consequence, we get information about the convergence rate of Lasserre’s procedure for optimization of a polynomial subject to polynomial constraints

Jiawang Nie

Markus Schweighofer

CiteSeerX

On the Complexity of Putinar’s Positivstellensatz

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.241.3528

On the complexity of Putinar's Positivstellensatz

Abstract

Similar works

Full text

Available Versions

KOPS - The Institutional Repository of the University of Konstanz

Elsevier - Publisher Connector

HAL-Rennes 1

CiteSeerX