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Decomposition of arrow type positive semidefinite matrices with application to topology optimization

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    SYSNO ASEP0532970
    Document TypeJ - Journal Article
    R&D Document TypeJournal Article
    Subsidiary JČlánek ve WOS
    TitleDecomposition of arrow type positive semidefinite matrices with application to topology optimization
    Author(s) Kočvara, Michal (UTIA-B) RID, ORCID
    Number of authors1
    Source TitleMathematical Programming. - : Springer - ISSN 0025-5610
    Roč. 190, 1-2 (2021), s. 105-134
    Number of pages30 s.
    Publication formPrint - P
    Languageeng - English
    CountryNL - Netherlands
    Keywordssemidefinite optimization ; positive semidefinite matrices ; chordal graphs ; domain decomposition ; topology optimization
    Subject RIVBA - General Mathematics
    OECD categoryPure mathematics
    Method of publishingOpen access
    Institutional supportUTIA-B - RVO:67985556
    UT WOS000539869200001
    EID SCOPUS85086374125
    DOI10.1007/s10107-020-01526-w
    AnnotationDecomposition of large matrix inequalities for matrices with chordal sparsity graph has been recently used by Kojima et al. (Math Program 129(1):33–68, 2011) to reduce problem size of large scale semidefinite optimization (SDO) problems and thus increase efficiency of standard SDO software. A by-product of such a decomposition is the introduction of new dense small-size matrix variables. We will show that for arrow type matrices satisfying suitable assumptions, the additional matrix variables have rank one and can thus be replaced by vector variables of the same dimensions. This leads to significant improvement in efficiency of standard SDO software. We will apply this idea to the problem of topology optimization formulated as a large scale linear semidefinite optimization problem. Numerical examples will demonstrate tremendous speed-up in the solution of the decomposed problems, as compared to the original large scale problem. In our numerical example the decomposed problems exhibit linear growth in complexity, compared to the more than cubic growth in the original problem formulation. We will also give a connection of our approach to the standard theory of domain decomposition and show that the additional vector variables are outcomes of the corresponding discrete Steklov–Poincaré operators.
    WorkplaceInstitute of Information Theory and Automation
    ContactMarkéta Votavová, votavova@utia.cas.cz, Tel.: 266 052 201.
    Year of Publishing2022
    Electronic addresshttps://link.springer.com/article/10.1007/s10107-020-01526-w
Number of the records: 1  

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