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PENNON

A Generalized Augmented Lagrangian Method for Semidefinite Programming
  • Michal Kočvara
  • Michael Stingl
Part of the Applied Optimization book series (APOP, volume 82)

Abstract

This article describes a generalization of the PBM method by Ben-Tal and Zibulevsky to convex semidefinite programming problems. The algorithm used is a generalized version of the Augmented Lagrangian method. We present details of this algorithm as implemented in a new code PENNON. The code can also solve second-order conic programming (SOCP) problems, as well as problems with a mixture of SDP, SOCP and NLP constraints. Results of extensive numerical tests and comparison with other SDP codes are presented.

Keywords

semidefinite programming cone programming method of augmented Lagrangians 

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References

  1. [1]
    A. Ben-Tal, F. Jarre, M. Kočvara, A. Nemirovski, and J. Zowe. Optimal design of trusses under a nonconvex global buckling constraint. Optimization and Engineering, 1:189–213, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    A. Ben-Tal, M. Kočvara, A. Nemirovski, and J. Zowe. Free material design via semidefinite programming. The multi-load case with contact conditions. SIAM J. Optimization,9:813–832, 1997.CrossRefGoogle Scholar
  3. [3]
    A. Ben-Tal and M. Zibulevsky. Penalty/barrier multiplier methods for convex programming problems. SIAM J. Optimization, 7:347–366, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    B. Borchers. CSDP, a C library for semidefinite programming. Op-timization Methods and Software,11:613–623, 1999. Available at http://www.nmt.edu/~borchers/ MathSciNetCrossRefGoogle Scholar
  5. [5]
    B. Borchers. SDPLIB 1.2, a library of semidefinite programming test problems. Optimization Methods and Software, 11 & 12:683–690, 1999. Available at http://www.nmt.edu/~borchers/ MathSciNetCrossRefGoogle Scholar
  6. [6]
    K. Fujisawa, M. Kojima, and K. Nakata. Exploiting sparsity in primal-dual interior-point method for semidefinite programming. Mathematical Programming, 79:235–253, 1997.MathSciNetzbMATHGoogle Scholar
  7. [7]
    H.R.E.M. Hörnlein, M. Kočvara, and R. Werner. Material optimization: Bridging the gap between conceptual and preliminary design. Aerospace Science and Technology, 2001. In print.Google Scholar
  8. [8]
    F. Jarre. An interior method for nonconvex semidefinite programs. Optimization and Engineering, 1:347–372, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    M. Kočvara. On the modelling and solving of the truss design problem with global stability constraints. Struct. Multidisc. Optimization, 2001. In print.Google Scholar
  10. [10]
    H. Mittelmann. Benchmarks for optimization software. Available at http://plato.la.asu.edu/bench.html
  11. [11]
    E. Ng and B. W. Peyton. Block sparse cholesky algorithms on advanced uniprocessor computers. SIAM J. Scientific Computing,14:1034–1056, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    G. Pataki and S. Schieta. The DIMACS library of mixed semidefinite-quadratic-linear problems. Available at http://dimacs.rutgers.edu/challenges/seventh/instances
  13. [13]
    M. Stingl. Konvexe semidefinite programmierung. Diploma Thesis, Institute of Applied Mathematics, University of Erlangen, 1999.Google Scholar
  14. [14]
    J. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 11 & 12:625–653, 1999. Available at http://fewcal.kub.nl/sturm/.MathSciNetCrossRefGoogle Scholar
  15. [15]
    R.H. Tütütcü, K.C. Toh, and M.J. Todd. SDPT3 — A MATLAB software package for semidefinite-quadratic-linear programming, Version 3.0. Available at http://www.orie.cornell.edu/~miketodd/todd.html School of Operations Research and Industrial Engineering, Cornell University, 2001.Google Scholar
  16. [16]
    J. Zowe, M. Kočvara, and M. Bendsøe. Free material optimization via mathematical programming. Mathematical Programming, Series B, 79:445–466, 1997.zbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers B.V. 2003

Authors and Affiliations

  • Michal Kočvara
    • 1
  • Michael Stingl
    • 1
  1. 1.Institute of Applied MathematicsUniversity of ErlangenErlangenGermany

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