A method for weighted projections to the positive definite cone

Type
Article

Authors
Valkonen, Tuomo

KAUST Grant Number
KUK-I1-007-43

Online Publication Date
2014-06-24

Print Publication Date
2015-10-03

Date
2014-06-24

Abstract
© 2014 Taylor & Francis. We study the numerical solution of the problem (Formula presented.) , where (Formula presented.) is a symmetric square matrix, and (Formula presented.) is a linear operator, such that (Formula presented.) is invertible. With (Formula presented.) the desired fractional duality gap, and (Formula presented.) the condition number of (Formula presented.) , we prove (Formula presented.) iteration complexity for a simple primal-dual interior point method directly based on those for linear programs with semi-definite constraints. We do not, however, require the numerically expensive scalings inherent in these methods to force fast convergence. For low-dimensional problems (Formula presented.), our numerical experiments indicate excellent performance and only a very slowly growing dependence of the convergence rate on (Formula presented.). While our algorithm requires somewhat more iterations than existing interior point methods, the iterations are cheaper. This gives better computational times.

Citation
Valkonen T (2014) A method for weighted projections to the positive definite cone. Optimization 64: 2253–2275. Available: http://dx.doi.org/10.1080/02331934.2014.929680.

Acknowledgements
This research was financially supported by the SFB research programme F32 ‘Mathematical Optimization and Applications in Biomedical Sciences’ of the Austrian Science Fund (FWF), and by the King Abdullah University of Science and Technology (KAUST) Award No. KUK-I1-007-43.

Publisher
Informa UK Limited

Journal
Optimization

DOI
10.1080/02331934.2014.929680

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