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Mathematical Programming Computation

, Volume 2, Issue 3–4, pp 167–201 | Cite as

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

  • Martin S. AndersenEmail author
  • Joachim Dahl
  • Lieven Vandenberghe
Open Access
Full Length Paper

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.

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 

Notes

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.

References

  1. 1.
    Amestoy P., Davis T., Duff I.: An approximate minimum degree ordering. SIAM J. Matrix Anal. Appl. 17(4), 886–905 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alizadeh F., Goldfarb D.: Second-order cone programming. Math. Program. Ser. B 95, 3–51 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Borchers B.: CSDP, a C library for semidefinite programming. Optim. Methods Softw. 11(1), 613–623 (1999)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Borchers B.: SDPLIB 1.2, a library of semidefinite programming test problems. Optim. Methods Soft. 11(1), 683–690 (1999)CrossRefMathSciNetGoogle Scholar
  7. 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. 8.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. Analysis, Algorithms, and Engineering Applications. Society for Pure and Applied Mathematics (2001)Google Scholar
  9. 9.
    Burer S.: Semidefinite programming in the space of partial positive semidefinite matrices. SIAM J. Optim. 14(1), 139–172 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004). http://www.stanford.edu/~boyd/cvxbook
  11. 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 SoftwareGoogle Scholar
  12. 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)CrossRefGoogle Scholar
  13. 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)Google Scholar
  14. 14.
    Dahl, J., Vandenberghe, L.: CVXOPT: A Python Package for Convex Optimization. http://abel.ee.ucla.edu/cvxopt (2008)
  15. 15.
    Dahl, J., Vandenberghe, L.: CHOMPACK: Chordal Matrix Package. http://abel.ee.ucla.edu/chompack (2009)
  16. 16.
    Dahl J., Vandenberghe L., Roychowdhury V.: Covariance selection for non-chordal graphs via chordal embedding. Optim. Methods Softw. 23(4), 501–520 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 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)MathSciNetGoogle Scholar
  20. 20.
    Fourer R., Mehrotra S.: Solving symmetric indefinite systems in an interior-point approach for linear programming. Math. Program. 62, 15–39 (1993)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming (web page and software). http://stanford.edu/~boyd/cvx (2007)
  22. 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)Google Scholar
  23. 23.
    George A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10(2), 345–363 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Goldfarb D., Iyengar G.: Robust convex quadratically constrained programs. Math. Program. Ser. B 97, 495–515 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Hauser R.A., Güler O.: Self-scaled barrier functions on symmetric cones and their classification. Found. Comput. Math. 2, 121–143 (2002)zbMATHMathSciNetGoogle Scholar
  27. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Johnson, D., Pataki, G., Alizadeh, F.: Seventh DIMACS implementation challenge: Semidefinite and related problems (2000). http://dimacs.rutgers.edu/Challenges/Seventh
  29. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Lauritzen S.L.: Graphical Models. Oxford University Press, Oxford (1996)Google Scholar
  32. 32.
    Löfberg, J.: YALMIP : A Toolbox for Modeling and Optimization in MATLAB (2004)Google Scholar
  33. 33.
    Löfberg, J.: YALMIP : A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004)Google Scholar
  34. 34.
    Monteiro R.D.C.: Primal–dual path following algorithms for semidefinite programming. SIAM J. Optim. 7, 663–678 (1995)CrossRefMathSciNetGoogle Scholar
  35. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Nesterov, Yu.: Nonsymmetric potential-reduction methods for general cones. Technical Report 2006/34, CORE Discussion Paper, Université catholique de Louvain (2006)Google Scholar
  37. 37.
    Nesterov, Yu.: Towards nonsymmetric conic optimization. Technical Report 2006/28, CORE Discussion Paper, Université catholique de Louvain (2006)Google Scholar
  38. 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)zbMATHCrossRefGoogle Scholar
  39. 39.
    Nesterov, Yu., Nemirovskii, A.: Interior-point polynomial methods in convex programming. Studies in Applied Mathematics, vol. 13. SIAM, Philadelphia (1994)Google Scholar
  40. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Nesterov Yu.E., Todd M.J.: Primal–dual interior-point methods for self-scaled cones. SIAM J. Optim. 8(2), 324–364 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization. Society for Industrial and Applied Mathematics (2001)Google Scholar
  43. 43.
    Rose D.J.: Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32, 597–609 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Sturm J.F.: Using SEDUMI 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)CrossRefMathSciNetGoogle Scholar
  46. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Sturm J.F.: Avoiding numerical cancellation in the interior point method for solving semidefinite programs. Math. Program. Ser. B 95, 219–247 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  48. 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/0412009v1Google Scholar
  49. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  50. 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)zbMATHCrossRefGoogle Scholar
  51. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Wermuth N.: Linear recursive equations, covariance selection, and path analysis. J. Am Stat. Assoc. 75(372), 963–972 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  54. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Wright S.J.: Primal–Dual Interior-Point Methods. SIAM, Philadelphia (1997)zbMATHGoogle Scholar
  56. 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)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  • Martin S. Andersen
    • 1
    Email author
  • Joachim Dahl
    • 2
  • Lieven Vandenberghe
    • 1
  1. 1.Electrical Engineering DepartmentUniversity of CaliforniaLos AngelesUSA
  2. 2.MOSEK ApSCopenhagen ØDenmark

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