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International Conference on Parallel Processing and Applied Mathematics

PPAM 2005: Parallel Processing and Applied Mathematics pp 513–525Cite as

Direct Solution of Linear Systems of Size 109 Arising in Optimization with Interior Point Methods

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 3911)

Abstract

Solution methods for very large scale optimization problems are addressed in this paper. Interior point methods are demonstrated to provide unequalled efficiency in this context. They need a small (and predictable) number of iterations to solve a problem. A single iteration of interior point method requires the solution of indefinite system of equations. This system is regularized to guarantee the existence of triangular decomposition. Hence the well-understood parallel computing techniques developed for positive definite matrices can be extended to this class of indefinite matrices. A parallel implementation of an interior point method is described in this paper. It uses object-oriented programming techniques and allows for exploiting different block-structures of matrices. Our implementation outperforms the industry-standard optimizer, shows very good parallel efficiency on massively parallel architecture and solves problems of unprecedented sizes reaching 109 variables.

Keywords

  • Portfolio Selection
  • Interior Point Method
  • Jacobian Matrice
  • Direct Solution
  • Interior Point Algorithm

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Supported by the Engineering and Physical Sciences Research Council of UK, EPSRC grant GR/R99683/01.

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  • DOI: 10.1007/11752578_62
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Author information

Authors and Affiliations

  1. School of Mathematics, University of Edinburgh, JCMB, King’s Buildings, Edinburgh, EH9 3JZ, UK

    Jacek Gondzio & Andreas Grothey

Authors
  1. Jacek Gondzio
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  2. Andreas Grothey
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Editor information

Editors and Affiliations

  1. Institute of Computational and Information Sciences, Czestochowa University of Technology, Poland

    Roman Wyrzykowski

  2. Computer Science Department,, University of Tennessee, 37996-3450, Knoxville, TN, USA

    Jack Dongarra

  3. Poznan Supercomputing and Networking Center, Poland

    Norbert Meyer

  4. Informatics & Mathematical Modeling, Technical University of Denmark, 2800, Lyngby, DK, Denmark

    Jerzy Waśniewski

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Gondzio, J., Grothey, A. (2006). Direct Solution of Linear Systems of Size 109 Arising in Optimization with Interior Point Methods. In: Wyrzykowski, R., Dongarra, J., Meyer, N., Waśniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2005. Lecture Notes in Computer Science, vol 3911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11752578_62

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  • DOI: https://doi.org/10.1007/11752578_62

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-34141-3

  • Online ISBN: 978-3-540-34142-0

  • eBook Packages: Computer ScienceComputer Science (R0)