Skip to main content

Advertisement

SpringerLink
Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones
Download PDF
Download PDF
  • Full Length Paper
  • Open Access
  • Published: 26 August 2010

Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones

  • Martin S. Andersen1,
  • Joachim Dahl2 &
  • Lieven Vandenberghe1 

Mathematical Programming Computation volume 2, pages 167–201 (2010)Cite this article

  • 1253 Accesses

  • 39 Citations

  • Metrics details

Abstract

We describe an implementation of nonsymmetric interior-point methods for linear cone programs defined by two types of matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity pattern and its dual cone, the cone of chordal sparse matrices that have a positive semidefinite completion. The implementation takes advantage of fast recursive algorithms for evaluating the function values and derivatives of the logarithmic barrier functions for these cones. We present experimental results of two implementations, one of which is based on an augmented system approach, and a comparison with publicly available interior-point solvers for semidefinite programming.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Amestoy P., Davis T., Duff I.: An approximate minimum degree ordering. SIAM J. Matrix Anal. Appl. 17(4), 886–905 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alizadeh F., Goldfarb D.: Second-order cone programming. Math. Program. Ser. B 95, 3–51 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alizadeh F., Haeberly J.-P.A., Overton M.L.: Primal–dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results. SIAM J. Optim. 8(3), 746–768 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barrett W.W., Johnson C.R., Lundquist M.: Determinantal formulation for matrix completions associated with chordal graphs. Linear Algebra Appl. 121, 265–289 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Borchers B.: CSDP, a C library for semidefinite programming. Optim. Methods Softw. 11(1), 613–623 (1999)

    Article  MathSciNet  Google Scholar 

  6. Borchers B.: SDPLIB 1.2, a library of semidefinite programming test problems. Optim. Methods Soft. 11(1), 683–690 (1999)

    Article  MathSciNet  Google Scholar 

  7. Blair J.R.S., Peyton B.: An introduction to chordal graphs and clique trees. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds) Graph Theory and Sparse Matrix Computation, Springer, Berlin (1993)

    Google Scholar 

  8. Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. Analysis, Algorithms, and Engineering Applications. Society for Pure and Applied Mathematics (2001)

  9. Burer S.: Semidefinite programming in the space of partial positive semidefinite matrices. SIAM J. Optim. 14(1), 139–172 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004). http://www.stanford.edu/~boyd/cvxbook

  11. Benson, S.J., Ye, Y.: DSDP5: Software for semidefinite programming. Technical Report ANL/MCS-P1289-0905, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, September 2005. Submitted to ACM Transactions on Mathematical Software

  12. Chen Y., Davis T.A., Hager W.W., Rajamanickam S.: Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate. ACM Trans. Math. Softw. 35(3), 1–14 (2008)

    Article  Google Scholar 

  13. Davis, T.A.: The University of Florida Sparse Matrix Collection. Technical report, Department of Computer and Information Science and Engineering, University of Florida (2009)

  14. Dahl, J., Vandenberghe, L.: CVXOPT: A Python Package for Convex Optimization. http://abel.ee.ucla.edu/cvxopt (2008)

  15. Dahl, J., Vandenberghe, L.: CHOMPACK: Chordal Matrix Package. http://abel.ee.ucla.edu/chompack (2009)

  16. Dahl J., Vandenberghe L., Roychowdhury V.: Covariance selection for non-chordal graphs via chordal embedding. Optim. Methods Softw. 23(4), 501–520 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. El Ghaoui L., Lebret H.: Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 18(4), 1035–1064 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. 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)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fujisawa K., Kojima M., Nakata K.: Exploiting sparsity in primal–dual interior-point methods for semidefinite programming. Math. Program. 79(1–3), 235–253 (1997)

    MathSciNet  Google Scholar 

  20. Fourer R., Mehrotra S.: Solving symmetric indefinite systems in an interior-point approach for linear programming. Math. Program. 62, 15–39 (1993)

    Article  MathSciNet  Google Scholar 

  21. Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming (web page and software). http://stanford.edu/~boyd/cvx (2007)

  22. Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs. In: Blondel, V., Boyd, S., Kimura, H. (eds.) Recent Advances in Learning and Control (a tribute to M. Vidyasagar). Springer, Berlin (2008)

  23. George A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10(2), 345–363 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  24. Goldfarb D., Iyengar G.: Robust convex quadratically constrained programs. Math. Program. Ser. B 97, 495–515 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Grone R., Johnson C.R., Sá E.M., Wolkowicz H.: Positive definite completions of partial Hermitian matrices. Linear Algebra Appl. 58, 109–124 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hauser R.A., Güler O.: Self-scaled barrier functions on symmetric cones and their classification. Found. Comput. Math. 2, 121–143 (2002)

    MATH  MathSciNet  Google Scholar 

  27. Helmberg C., Rendl F., Vanderbei R.J., Wolkowicz H.: An interior-point method for semidefinite programming. SIAM J. Optim. 6(2), 342–361 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. Johnson, D., Pataki, G., Alizadeh, F.: Seventh DIMACS implementation challenge: Semidefinite and related problems (2000). http://dimacs.rutgers.edu/Challenges/Seventh

  29. Kobayashi K., Kim S., Kojima M.: Correlative sparsity in primal–dual interior-point methods for LP, SDP, and SOCP. Appl. Math. Optim. 58(1), 69–88 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kojima M., Shindoh S., Hara S.: Interior-point methods for the monotone linear complementarity problem in symmetric matrices. SIAM J. Optim. 7, 86–125 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lauritzen S.L.: Graphical Models. Oxford University Press, Oxford (1996)

    Google Scholar 

  32. Löfberg, J.: YALMIP : A Toolbox for Modeling and Optimization in MATLAB (2004)

  33. Löfberg, J.: YALMIP : A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004)

  34. Monteiro R.D.C.: Primal–dual path following algorithms for semidefinite programming. SIAM J. Optim. 7, 663–678 (1995)

    Article  MathSciNet  Google Scholar 

  35. Monteiro R.D.C.: Polynomial convergence of primal–dual algorithms for semidefinite programming based on Monteiro and Zhang family of directions. SIAM J. Optim. 8(3), 797–812 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  36. Nesterov, Yu.: Nonsymmetric potential-reduction methods for general cones. Technical Report 2006/34, CORE Discussion Paper, Université catholique de Louvain (2006)

  37. Nesterov, Yu.: Towards nonsymmetric conic optimization. Technical Report 2006/28, CORE Discussion Paper, Université catholique de Louvain (2006)

  38. Nakata K., Fujitsawa K., Fukuda M., Kojima M., Murota K.: Exploiting sparsity in semidefinite programming via matrix completion. II. Implementation and numerical details. Math. Program. Ser. B 95, 303–327 (2003)

    Article  MATH  Google Scholar 

  39. Nesterov, Yu., Nemirovskii, A.: Interior-point polynomial methods in convex programming. Studies in Applied Mathematics, vol. 13. SIAM, Philadelphia (1994)

  40. Nesterov Yu.E., Todd M.J.: Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res. 22(1), 1–42 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  41. Nesterov Yu.E., Todd M.J.: Primal–dual interior-point methods for self-scaled cones. SIAM J. Optim. 8(2), 324–364 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  42. Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization. Society for Industrial and Applied Mathematics (2001)

  43. Rose D.J.: Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32, 597–609 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  44. Rose D.J., Tarjan R.E., Lueker G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  45. Sturm J.F.: Using SEDUMI 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)

    Article  MathSciNet  Google Scholar 

  46. Sturm J.F.: Implementation of interior point methods for mixed semidefinite and second order cone optimization problems. Optim. Methods Softw. 17(6), 1105–1154 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  47. Sturm J.F.: Avoiding numerical cancellation in the interior point method for solving semidefinite programs. Math. Program. Ser. B 95, 219–247 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  48. Srijuntongsiri, G., Vavasis, S.A.: A fully sparse implementation of a primal–dual interior-point potential reduction method for semidefinite programming (2004). Available at arXiv: arXiv:cs/0412009v1

  49. Todd M.J., Toh K.C., Tütüncü R.H.: On the Nesterov–Todd direction in semidefinite programming. SIAM J. Optim. 8(3), 769–796 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  50. Tütüncü R.H., Toh K.C., Todd M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. Ser. B 95, 189–217 (2003)

    Article  MATH  Google Scholar 

  51. Tarjan R.E., Yannakakis M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  52. Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  53. Wermuth N.: Linear recursive equations, covariance selection, and path analysis. J. Am Stat. Assoc. 75(372), 963–972 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  54. Waki H., Kim S., Kojima M., Muramatsu M.: Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17(1), 218–241 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  55. Wright S.J.: Primal–Dual Interior-Point Methods. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  56. Yamashita M., Fujisawa K., Kojima M.: Implementation and evaluation of SDPA 6.0 (Semidefinite Programming Algorithm 6.0). Optim. Methods Softw. 18(4), 491–505 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Authors and Affiliations

  1. Electrical Engineering Department, University of California, Los Angeles, USA

    Martin S. Andersen & Lieven Vandenberghe

  2. MOSEK ApS, Fruebjergvej 3, 2100, Copenhagen Ø, Denmark

    Joachim Dahl

Authors
  1. Martin S. Andersen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joachim Dahl
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lieven Vandenberghe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin S. Andersen.

Additional information

The authors’ (Andersen and Vandenberghe) research was supported in part by NSF grants ECS-0524663 and ECCS-0824003.

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Andersen, M.S., Dahl, J. & Vandenberghe, L. Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones. Math. Prog. Comp. 2, 167–201 (2010). https://doi.org/10.1007/s12532-010-0016-2

Download citation

  • Received: 31 July 2009

  • Accepted: 02 August 2010

  • Published: 26 August 2010

  • Issue Date: December 2010

  • DOI: https://doi.org/10.1007/s12532-010-0016-2

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (2000)

  • 90-08 Mathematical Programming - computational methods
  • 90C06 Mathematical Programming - large-scale
  • 90C22 Mathematical Programming - semidefinite programing
  • 90C25 Mathematical Programming - convex programming
  • 90C51 Mathematical Programming - interior-point methods
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Over 10 million scientific documents at your fingertips

Switch Edition
  • Academic Edition
  • Corporate Edition
  • Home
  • Impressum
  • Legal information
  • Privacy statement
  • California Privacy Statement
  • How we use cookies
  • Manage cookies/Do not sell my data
  • Accessibility
  • FAQ
  • Contact us
  • Affiliate program

Not affiliated

Springer Nature

© 2023 Springer Nature Switzerland AG. Part of Springer Nature.