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Theses Year : 2022

Practical polynomial optimization through positivity certificates with and without denominators

Optimisation polynomiale effective via des certificats de positivité avec ou sans dénominateurs

Ngoc Hoang Anh Mai
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Abstract

Positivity certificates or Positivstellensätze provide representations of polynomials positive on basic semialgebraic sets, i.e., sets defined by finitely many polynomial inequalities. The famous Putinar's Positivstellensatz states that every positive polynomial on a basic closed semialgebraic set S can be written as a linear weighted combination of the polynomials describing S, under a certain condition on S slightly stronger than compactness. When written in this it becomes obvious that the polynomial is positive on S, and therefore this alternative description provides a certificate of positivity on S. Moreover, as the polynomial weights involved in Putinar's Positivstellensatz are sums of squares (SOS), such Positivity certificates enable to design convex relaxations based on semidefinite pro- gramming to solve polynomial optimization problems (POPs) that arise in various real-life appli- cations, e.g., in management of energy networks and machine learning to cite a few. Originally developed by Lasserre, the hierarchy of semidefinite relaxations based on Putinar's Positivstellen- satz is called the Moment-SOS hierarchy. In this thesis, we provide polynomial optimization methods based on positivity certificates involving specific SOS weights, without or with denominators. The first part of this thesis focuses on methods without denominators, and based on Putinar's Positivstellensatz. The second part of this manuscript focuses on methods with denominators, based on positivity certificates due to Putinar and Vasilescu, Reznick, as well as Polya
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Dates and versions

tel-03843327 , version 1 (08-11-2022)
tel-03843327 , version 2 (26-01-2023)

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  • HAL Id : tel-03843327 , version 1

Cite

Ngoc Hoang Anh Mai. Practical polynomial optimization through positivity certificates with and without denominators. Automatic Control Engineering. UT3 : Université Toulouse 3 Paul Sabatier, 2022. English. ⟨NNT : ⟩. ⟨tel-03843327v1⟩
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