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Exploiting sparsity in primal-dual interior-point methods for semidefinite programming

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Abstract

The Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro and Nesterov-Todd search directions have been used in many primal-dual interior-point methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when the semidefinite program to be solved is large scale and sparse.

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References

  1. F. Alizadeh, J.-P.A. Haeberly and M.L. Overton, Primal-dual interior-point methods for semidefinite programming, Working Paper, 1994.

  2. F. Alizadeh, J.-P.A. Haeberly and M.L. Overton, Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results,SIAM Journal on Optimization, to appear.

  3. N. Brixius, F.A. Potra and R. Sheng, Solving semidefinite programming in mathematica, Reports on Computational Mathematics, No. 97/1996, Dept. of Mathematics, University of Iowa, 1996; Available at http://www.cs.uiowa.edu/brixius/sdp.html.

  4. S. Boyd, L.E. Ghaoui, E. Feron and V. Balakrishnan,Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, PA, 1994).

    MATH  Google Scholar 

  5. K. Fujisawa, M. Kojima and K. Nakata, SDPA (Semidefinite Programming Algorithm) — Userz’s Manual —, Tech. Rept. B-308, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo 152, Japan, 1995, revised 1996; Available via anonymous ftp at ftp.is.titech.ac.jp in pub/OpRes/software/SDPA.

  6. C. Helmberg, F. Rendl, R.J. Vanderbei and H. Wolkowicz, An interior-point method for semidefinite programming,SIAM Journal on Optimization 6 (1996) 342–361.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Kojima, M. Shida and S. Shindoh, Local convergence of predictor-corrector infeasible-interior-point algorithms for SDPs and SDLCPs,Mathematical Programming, to appear.

  8. M. Kojima, S. Shindoh and S. Hara, Interior-point methods for the monotone semidefinite linear complementarity problems,SIAM Journal on Optimization 7 (1997) 86–125.

    Article  MATH  MathSciNet  Google Scholar 

  9. C.-J. Lin and R. Saigal, A predictor-corrector method for semi-definite linear programming, Working paper, Dept. of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, MI, 1995.

  10. C.-J. Lin and R. Saigal, Predictor corrector methods for semidefinite programming, Informs Atlanta Fall Meeting, 1996.

  11. Z.-Q. Luo, J.F. Sturm and S. Zhang, Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming,SIAM Journal of Optimization, to appear.

  12. R.D.C. Monteiro, Primal—dual path following algorithms for semidefinite programming,SIAM Journal on Optimization, to appear.

  13. Yu.E. Nesterov and M.J. Todd, Self-scaled cones and interior-point methods in nonlinear programming, Working Paper, CORE, Catholic University of Louvain, Louvain-la-Neuve, Belgium, 1994; also in:Mathematics of Operations Research, to appear.

  14. Yu.E. Nesterov and M.J. Todd, Primal-dual interior-point methods for self-scaled cones,SIAM Journal on Optimization, to appear.

  15. S. Poljak, F. Rendl and H. Wolkowicz, A recipe for semidefinite relaxation for (0,1) quadratic programming,Journal of Global Optimization 7 (1995) 51–73.

    Article  MATH  MathSciNet  Google Scholar 

  16. F.A. Potra and R. Sheng, Superlinear convergence of infeasible-interior-point algorithms for semidefinite programming, Dept. of Mathematics, University of Iowa, Iowa City, IA, 1996.

    Google Scholar 

  17. M.J. Todd, K.C. Toh and R.H. Tütüncü, On the Nesterov-Todd direction in semidefinite programming, Tech. Rept., School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 1996.

    Google Scholar 

  18. K.C. Toh, M.J. Todd and R.H. Tütüncü, SDPT3 — A MATLAB software package for semidefinite programming, Dept. of Mathematics, National University of Singapore, Singapore, 1996; Available at http://www.math.nus.sg/mattohkc/index.html.

  19. Y. Zhang, On extending primal-dual interior-point algorithms from linear programming to semidefinite programming,SIAM Journal on Optimization, to appear.

  20. Q. Zhao, S.E. Karisch, F. Rendl and H. Wolkowicz, Semidefinite programming relaxations for the quadratic assignment problem, CORR Rept. 95-27, Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, 1996.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, 152, Tokyo, Japan

    Katsuki Fujisawa, Masakazu Kojima & Kazuhide Nakata

Authors
  1. Katsuki Fujisawa
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  2. Masakazu Kojima
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  3. Kazuhide Nakata
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Fujisawa, K., Kojima, M. & Nakata, K. Exploiting sparsity in primal-dual interior-point methods for semidefinite programming. Mathematical Programming 79, 235–253 (1997). https://doi.org/10.1007/BF02614319

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  • DOI: https://doi.org/10.1007/BF02614319

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