Solving SDP's in Non-commutative Algebras Part I: The Dual-Scaling Algorithm
AbstractSemidefinite programming (SDP) may be viewed as an extension of linear programming (LP), and most interior point methods (IPM s) for LP can be extended to solve SDP problems.However, it is far more difficult to exploit data structures (especially sparsity) in the SDP case.In this paper we will look at the data structure where the SDP data matrices lie in a low dimensional matrix algebra.This data structure occurs in several applications, including the lower bounding of the stability number in certain graphs and the crossing number in complete bipartite graphs.We will show that one can reduce the linear algebra involved in an iteration of an IPM to involve matrices of the size of the dimension of the matrix algebra only.In other words, the original sizes of the data matrices do not appear in the computational complexity bound.In particular, we will work out the details for the dual scaling algorithm, since a dual method is most suitable for the types of applications we have in mind.
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Bibliographic InfoPaper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 2005-17.
Date of creation: 2005
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semidefinite programming; matrix algebras; dual scaling algorithm; exploiting data structure;
Find related papers by JEL classification:
- C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
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- Klerk, E. de, 1997. "Interior point methods for semidefinite programming," Open Access publications from Tilburg University urn:nbn:nl:ui:12-226108, Tilburg University.
- GOEMANS, Michel & RENDL, Franz, 1999. "Semidefinite programs and association schemes," CORE Discussion Papers 1999062, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Klerk, E. de & Pasechnik, D.V., 2005. "A Note on the Stability Number of an Orthogonality Graph," Discussion Paper 2005-66, Tilburg University, Center for Economic Research.
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