On the solution of large-scale SDP problems by the modified barrier method using iterative solvers |
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Authors: | Michal Ko?vara Michael Stingl |
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Institution: | (1) Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 18208 Praha 8, Czech Republic;(2) Faculty of Electrical Engineering, Czech Technical University, Technická 2, 166 27 Prague, Czech Republic;(3) Institute of Applied Mathematics, University of Erlangen, Martensstr. 3, 91058 Erlangen, Germany |
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Abstract: | The limiting factors of second-order methods for large-scale semidefinite optimization are the storage and factorization of
the Newton matrix. For a particular algorithm based on the modified barrier method, we propose to use iterative solvers instead
of the routinely used direct factorization techniques. The preconditioned conjugate gradient method proves to be a viable
alternative for problems with a large number of variables and modest size of the constrained matrix. We further propose to
avoid explicit calculation of the Newton matrix either by an implicit scheme in the matrix–vector product or using a finite-difference
formula. This leads to huge savings in memory requirements and, for certain problems, to further speed-up of the algorithm.
Dedicated to the memory of Jos Sturm. |
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Keywords: | 90C22 (primary) 65F10 (secondary) |
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