EXACT ALGORITHMS FOR LINEAR MATRIX INEQUALITIES

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F16%3A00304330" target="_blank" >RIV/68407700:21230/16:00304330 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1137/15M1036543" target="_blank" >http://dx.doi.org/10.1137/15M1036543</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1137/15M1036543" target="_blank" >10.1137/15M1036543</a>

Jazyk výsledku

angličtina

Název v původním jazyce

EXACT ALGORITHMS FOR LINEAR MATRIX INEQUALITIES

Popis výsledku v původním jazyce

Let A(x) = A0 +x1A1+xnAn be a linear matrix, or pencil, generated by given symmetric matrices A0;A1;An of size m with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron, described by a linear matrix inequality. We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semide finite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a multilinear Bezout bound of d. When m (resp., n) is fixed, such a bound, and hence the complexity, is polynomial in n (resp., m). We conclude by providing results of experiments showing practical improvements with respect to state-of-The-Art computer algebra algorithms.

Název v anglickém jazyce

EXACT ALGORITHMS FOR LINEAR MATRIX INEQUALITIES

Popis výsledku anglicky

Let A(x) = A0 +x1A1+xnAn be a linear matrix, or pencil, generated by given symmetric matrices A0;A1;An of size m with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron, described by a linear matrix inequality. We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semide finite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a multilinear Bezout bound of d. When m (resp., n) is fixed, such a bound, and hence the complexity, is polynomial in n (resp., m). We conclude by providing results of experiments showing practical improvements with respect to state-of-The-Art computer algebra algorithms.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

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Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2016

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

SIAM JOURNAL ON OPTIMIZATION

ISSN

1052-6234

e-ISSN

—

Svazek periodika

26

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

28

Strana od-do

2512-2539

Kód UT WoS článku

000391853600021

EID výsledku v databázi Scopus

2-s2.0-85007170438