Mathematical Programming

, Volume 95, Issue 2, pp 219–247

Avoiding numerical cancellation in the interior point method for solving semidefinite programs

  • Jos F. Sturm

DOI: 10.1007/s10107-002-0348-4

Cite this article as:
Sturm, J. Math. Program., Ser. B (2003) 95: 219. doi:10.1007/s10107-002-0348-4


 The matrix variables in a primal-dual pair of semidefinite programs are getting increasingly ill-conditioned as they approach a complementary solution. Multiplying the primal matrix variable with a vector from the eigenspace of the non-basic part will therefore result in heavy numerical cancellation. This effect is amplified by the scaling operation in interior point methods. A complete example illustrates these numerical issues. In order to avoid numerical problems in interior point methods, we propose to maintain the matrix variables in a Cholesky form. We discuss how the factors of the v-space Cholesky form can be updated after a main iteration of the interior point method with Nesterov-Todd scaling. An analogue for second order cone programming is also developed. Numerical results demonstrate the success of this approach.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Jos F. Sturm
    • 1
  1. 1.Department of Econometrics, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands, e-mail: j.f.sturm@uvt.nlNL

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