Hilbert's 17th problem and best dual bounds in quadratic minimization
type: Item
instance of: scholarly article
title: Hilbert's 17th problem and best dual bounds in quadratic minimization
zbMATH Open document ID: 0972.13017
DOI: 10.1007/BF02667043
author: Ch. Ferrier
published in: Cybernetics and Systems Analysis
publication date: 2000-05-08T00:00:00Z
review text: Positive semi-definite polynomials in \(\mathbb{R}[X]= \mathbb{R}[X_1, \dots, X_n]\) can always be written as sums of squares in the function field \(\mathbb{R}(X_1, \dots,X_n)\) (Hilbert's 17-th problem, solved by Artin), but frequently not in the polynomial ring. Many variations of this classsical problem have been studied, resulting in various Positivstellensätze and Nicht-Negativstellensätze. The present paper asks: Given a polynomial \(P\in\mathbb{R} [X]\), when does there exist some \(c\in\mathbb{R}\) such that \(P-c\) is a sum of squares? The question is clearly related to the optimisation problem: Determine \(\inf\{P(x)\mid x\in \mathbb{R}^n\}\). Inflating the number of variables, the author transforms this problem into an optimisation problem about quadratic polynomials. He proves that the above question has an affirmative answer if and only if the dual bound of the quadratic optimisation problem is finite. Variations for polynomial functions on subvarieties of \(\mathbb{R}^n\) or on basic closed semi-algebraic sets are also included. The paper has a large number of typos.
reviewed by: Niels Schwartz
Mathematics Subject Classification ID: 13J30
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 1418953
zbMATH Keywords: positive semi-definite polynomials
zbMATH Keywords: sums of squares
zbMATH Keywords: semi-algebraic sets
MaRDI profile type: Publication
cites work: Q3770650
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cites work: Q4698624
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full work available at URL: https://doi.org/10.1007/bf02667043
OpenAlex ID: W2001393186
description: scientific article; zbMATH DE number 1418953