0532970 - ÚTIA 2022 RIV NL eng J - Journal Article
Kočvara, Michal
Decomposition of arrow type positive semidefinite matrices with application to topology optimization.
Mathematical Programming. Roč. 190, 1-2 (2021), s. 105-134. ISSN 0025-5610. E-ISSN 1436-4646
Institutional support: RVO:67985556
Keywords : semidefinite optimization * positive semidefinite matrices * chordal graphs * domain decomposition * topology optimization
OECD category: Pure mathematics
Impact factor: 3.060, year: 2021
Method of publishing: Open access
Result website:
http://library.utia.cas.cz/separaty/2020/MTR/kocvara-0532970.pdf https://link.springer.com/article/10.1007/s10107-020-01526-w
DOI: https://doi.org/10.1007/s10107-020-01526-w

Decomposition 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.

Permanent Link: http://hdl.handle.net/11104/0311548