A conversion of an SDP having free variables into the standard form SDP

  • Kazuhiro Kobayashi
  • Kazuhide Nakata
  • Masakazu Kojima


This paper deals with a semidefinite program (SDP) having free variables, which often appears in practice. To apply the primal–dual interior-point method, we usually need to convert our SDP into the standard form having no free variables. One simple way of conversion is to represent each free variable as a difference of two nonnegative variables. But this conversion not only expands the size of the SDP to be solved but also yields some numerical difficulties which are caused by the non-existence of a primal–dual pair of interior-feasible solutions in the resulting standard form SDP and its dual. This paper proposes a new conversion method that eliminates all free variables. The resulting standard form SDP is smaller in its size, and it can be more stably solved in general because the SDP and its dual have interior-feasible solutions whenever the original primal–dual pair of SDPs have interior-feasible solutions. Effectiveness of the new conversion method applied to SDPs having free variables is reported in comparison to some other existing methods.


Semidefinite program Primal–dual interior-point method Equality constraint Standard form Conversion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahuja, R., Magnanti, T., Orlin, J.: Network Flows—Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993) Google Scholar
  2. 2.
    Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., Mckenney, A., Sorensen, D.: LAPACK Users’ Guide Third. SIAM, Philadelphia (1999) Google Scholar
  3. 3.
    Borchers, B.: SDPLIB 1.2, a library of semidefinite programming test problems. Optim. Methods Software 11&12, 683–690 (1999) MathSciNetGoogle Scholar
  4. 4.
    Fujisawa, K., Fukuda, M., Kojima, M., Nakata, K.: Numerical evaluation of SDPA (SemiDefinite Programming Algorithm). In: Frenk, H., Roos, K., Terlaky, T., Zhang, S. (eds.) High Performance Optimization, pp. 267–301. Kluwer Academic, Dordrecht (2000) Google Scholar
  5. 5.
    Fujisawa, K., Kojima, M., Nakata, K.: Exploiting sparsity in primal-dual interior-point methods for semidefinite programming, Math. Program. 79, 235–253 (1997) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Fukuda, M., Braams, B.J., Nakata, M., Overton, M.L., Percus, J.K., Yamashita, M., Zhao, Z.: Large-scale semidefinite programs in electronic structure calculation. Math. Program., to appear Google Scholar
  7. 7.
    Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion I: General framework. SIAM J. Optim. 11, 647–674 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Helmberg, C.: Semidefinite programming for combinatorial optimization. Habilitationsschrift. ZIP Report ZR-0034, TU Berlin, Konrad-Zuse-Zentrum, Berlin (2000). Available at
  9. 9.
    Helmberg, C., Rendl, F., Vanderbei, R.J., Wolkowicz, H.: An interior-point method for semidefinite programming. SIAM J. Optim. 6, 342–361 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kojima, M., Shindoh, S., Hara, S.: Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices. SIAM J. Optim. 7, 86–125 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lasserre, J.B.: Global optimization with polynomials and the problems of moments. SIAM J. Optim. 11, 796–817 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Monteiro, R.D.C.: Primal–dual path-following algorithms for semidefinite programming. SIAM J. Optim. 7, 663–678 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Nakata, K., Fujisawa, K., Fukuda, M., Kojima, M., Murota, K.: Exploiting sparsity in semidefinite programming via matrix completion II: Implementation and numerical results. Math. Program. 95, 303–327 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nakata, K., Yamashita, M., Fujisawa, K., Kojima, M.: A parallel primal–dual interior-point method for semidefinite programs using positive definite matrix completion. Parallel Comput. 32, 24–43 (2006) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Nakata, M., Nakatsuji, H., Ehara, M., Fukuda, M., Nakata, K., Fujisawa, K.: Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm. J. Chem. Phys. 114, 8282–8292 (2001) CrossRefGoogle Scholar
  16. 16.
    Nesterov, Yu.E., Todd, M.J.: Primal–dual interior-point methods for self-scaled cones. SIAM J. Optim. 8, 324–364 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Software 11&12, 625–653 (1999) MathSciNetGoogle Scholar
  18. 18.
    Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. 95, 189–217 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17, 218–242 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
  21. 21.
    Vanderbei, R.J., Carpenter, T.J.: Symmetric indefinite systems for interior point methods. Math. Program. 58, 1–32 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Yamashita, M., Fujisawa, K., Kojima, M.: Implementation and evaluation of SDPA6.0 (SemiDefinite Programming Algorithm 6.0). Optim. Methods Software 18, 491–505 (2003) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Kazuhiro Kobayashi
    • 1
  • Kazuhide Nakata
    • 2
  • Masakazu Kojima
    • 1
  1. 1.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of Industrial Engineering and ManagementTokyo Institute of TechnologyTokyoJapan

Personalised recommendations