Skip to main content
Log in

Interior-Point Algorithms for Semidefinite Programming Based on a Nonlinear Formulation

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Recently in Burer et al. (Mathematical Programming A, submitted), the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n × n matrix-valued function of a certain form into the positivity constraint on n scalar variables while keeping the number of variables unchanged. Based on this transformation, they proposed a first-order interior-point algorithm for solving a special class of linear semidefinite programs. In this paper, we extend this approach and apply the transformation to general linear semidefinite programs, producing nonlinear programs that have not only the n positivity constraints, but also n additional nonlinear inequality constraints. Despite this complication, the transformed problems still retain most of the desirable properties. We propose first-order and second-order interior-point algorithms for this type of nonlinear program and establish their global convergence. Computational results demonstrating the effectiveness of the first-order method are also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Benson and Y. Ye, “Approximating maximum stable set and minimum graph coloring problems with the positive semidefinite relaxation,” in Applications and Algorithms of Complementarity, M. Ferris and J. Pang (Eds.), Kluwer: Norwell, MA, 2000.

    Google Scholar 

  2. S. Benson, Y. Ye, and X. Zhang, “Solving large-scale sparse semidefinite programs for combinatorial optimization,” SIAM Journal on Optimization, vol. 10, pp. 443–461, 2000.

    Google Scholar 

  3. S. Burer and R.D.C. Monteiro, “A projected gradient algorithm for solving the maxcut SDP relaxation,” Optimization Methods and Software, vol. 15, pp. 175–200, 2001.

    Google Scholar 

  4. S. Burer and R.D.C. Monteiro, “A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization,” Working paper, School of ISyE, Georgia Tech, USA, March 2001. Also in Mathematical Programming, to appear.

    Google Scholar 

  5. S. Burer, R.D.C. Monteiro, and Y. Zhang, “Solving a class of semidefinite programs via nonlinear programming,” Mathematical Programming A, to appear. See also Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005, USA, Technical Report TR99-17, September 1999.

  6. C. Choi and Y. Ye, “Application of semidefinite programming to circuit partitioning,” in Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems, P.M. Pardalos (Ed.), Kluwer: Norwell, MA, 2000.

    Google Scholar 

  7. M.X. Goemans and D.P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” Journal of ACM, vol. 42, pp. 1115–1145, 1995.

    Google Scholar 

  8. C. Helmberg and F. Rendl, “A spectral bundle method for semidefinite programming,” SIAM Journal on Optimization, vol. 10, pp. 673–696, 2000.

    Google Scholar 

  9. D. Johnson and M. Trick, Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, AMS: Providence, RI, 1996.

    Google Scholar 

  10. L. Lovász, “On the Shannon capacity of a graph,” IEEE Transactions of Information Theory, vol. IT-25, no. 1, pp. 1–7, 1979.

    Google Scholar 

  11. R.D.C. Monteiro and J.S. Pang, “A potential reduction Newton method for constrained equations,” SIAM Journal on Optimization, vol. 9, pp. 729–754, 1999.

    Google Scholar 

  12. M. Peinado and S. Homer, “Design and performance of parallel and distributed approximation algorithms for maxcut,” Journal of Parallel and Distributed Computing, vol. 46, pp. 48–61, 1997.

    Google Scholar 

  13. R.J. Vanderbei and H. Yurttan Benson, “On formulating semidefinite programming problems as smooth convex nonlinear optimization problems,” Dept. of Operations Research and Financial Engineering, Princeton University, Princeton NJ, Technical Report ORFE 99-01, November 1999.

    Google Scholar 

  14. S. Vavasis, “A note on efficient computation of the gradient in semidefinite programming,” Working paper, Department of Computer Science, Cornell University, September 1999.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burer, S., Monteiro, R.D. & Zhang, Y. Interior-Point Algorithms for Semidefinite Programming Based on a Nonlinear Formulation. Computational Optimization and Applications 22, 49–79 (2002). https://doi.org/10.1023/A:1014834318702

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014834318702

Navigation