On verified numerical computations in convex programming

  • Christian JanssonEmail author


This survey contains recent developments for computing verified results of convex constrained optimization problems, with emphasis on applications. Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analysis, for linear programming, and for semidefinite programming. A discussion of important problem transformations to special types of convex problems and convex relaxations is included. The latter are important for handling and for reliability issues in global robust and combinatorial optimization. Some remarks on numerical experiences, including also large-scale and ill-posed problems, and software for verified computations concludes this survey.

Key words

linear programming semidefinite programming conic programming convex programming combinatorial optimization rounding errors ill-posed problems interval arithmetic branch-bound-and-cut 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. Alefeld and J. Herzberger, Introduction to Interval Computations. Academic Press, New York, 1983.zbMATHGoogle Scholar
  2. [2]
    F. Alizadeh and D. Glodfarb, Second-order cone programming. Math. Program.,95 (2003), 3–51.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    E.D. Andersen, C. Roos and T. Terlaky, A primal-dual interior-point method for conic quadratic optimization. Math. Programming,95 (2003), 249–277.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    H. Beeck, Linear programming with inexact data. Technical Report 7830, Abteilung Mathematik, TU München, 1978.Google Scholar
  5. [5]
    A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust semidefinite programming. Handbook of Semidefinite Programming, H. Wolkowicz, R. Saigal and L. Vandenberghe (eds.), Kluwer Academic Publishers, 2000.Google Scholar
  6. [6]
    A. Ben-Tal and A. Nemirovski, Robust convex optimization. Math. Operations Res.,23 (1998), 769–805.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPS-SIAM Series on Optimization, SIAM, Philadelphia, PA, 2001.zbMATHGoogle Scholar
  8. [8]
    S.J. Benson and Y. Ye, DSDP3: Dual scaling algorithm for general positive semidefinite programming. Technical Report Preprint ANL/MCS-P851-1000, Argonne National Labs, 2001.Google Scholar
  9. [9]
    M. Berz et al., COSY Infinity. Scholar
  10. [10]
    G.D. Birkhoff, Lattice Theory, revised edition. Am. Math. Soc. Colloquium Publications, Vol. 25, Am. Math. Soc., New York, 1948.Google Scholar
  11. [11]
    B. Borchers, CSDP, A C library for semidefinite programming. Optimization Methods and Software,11 (1999), 613–623.CrossRefMathSciNetGoogle Scholar
  12. [12]
    B. Borchers, SDPLIB 1.2, a library of semidefinite programming test problems. Optimization Methods and Software,11 (1999), 683–690.CrossRefMathSciNetGoogle Scholar
  13. [13]
    N. Bourbaki, Éléments de mathématique. XIII. 1 part: Les structures fondamentales de l’analyse, Livre VI: Intégration, Actualités scientifique et industrielles, 1952.Google Scholar
  14. [14]
    S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.Google Scholar
  15. [15]
    S. Burer and R.D.C. Monteiro, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Programming,95 (2003), 329–357.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    S. Burer, R.D.C. Monteiro and Y. Zhang, Solving a class of semidefinite programs via nonlinear programming. Math. Programming,93 (2002), 97–122.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    D. Chaykin, Verified Semidefinite Programming: Applications and the Software Package verified SDP. Ph.D. thesis, Technische Universität Hamburg-Harburg, 2009.Google Scholar
  18. [18]
    E. De Klerk, Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications. Dordrecht: Kluwer Academic Publishers, 2002.zbMATHGoogle Scholar
  19. [19]
    L. El Ghaoui, F. Oustry and H. Lebret, Robust Solutions to Uncertain Semidefinite Programs. SIAM J. Optim.,9 (1998), 33–52.zbMATHCrossRefGoogle Scholar
  20. [20]
    R.M. Freund, F. Ordóñez and K. Toh, Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems. Math. Programming,109 (2007), 445–475.zbMATHCrossRefGoogle Scholar
  21. [21]
    E.R. Hansen, Global Optimization Using Interval Analysis. Marcel Dekker, New York, 1992.zbMATHGoogle Scholar
  22. [22]
    C. Helmberg, SBmethoda C++ implementation of the spectral bundle method. Technical Report, Konrad-Zuse-Zentrum für Informationstechnik Berlin, 2000, Manual to Version 1.1, ZIB-Report ZR 00-35, Scholar
  23. [23]
    C. Helmberg, Semidefinite programming for combinatorial optimization (Habilitationsschrift). Technical Report ZIB ZR-00-34, Konrad-Zuse-Zentrum Berlin, TU Berlin, 2000.Google Scholar
  24. [24]
    C. Helmberg and K.C. Kiwiel, A spectral bundle method with bounds. Math. Programming,93 (2002), 173–194.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    ILOG CPLEX 7.1, User’s Manual. ILOG, France, 2001.Google Scholar
  26. [26]
    C. Jansson, A self-validating method for solving linear programming problems with interval input data, Computing Suppl.,6 (1988), 33–45.MathSciNetGoogle Scholar
  27. [27]
    C. Jansson, A rigorous lower bound for the optimal value of convex optimization problems. J. Global Optimization,28 (2004), 121–137.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    C. Jansson, Rigorous lower and upper bounds in linear programming. SIAM J. Optimization (SIOPT),14 (2004), 914–935.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    C. Jansson, VSDP: A MATLAB software package for verified semidefinite programming. NOLTA, 2006, 327–330.Google Scholar
  30. [30]
    C. Jansson, VSDP: Verified Semidefinite Programming, User’s Guide. 2006, http://www.BetaVersion0.1. Scholar
  31. [31]
    C. Jansson, Guaranteed accuracy for conic programming problems in vector lattices. 2007, arXiv:0707.4366v1, Scholar
  32. [32]
    C. Jansson, D. Chaykin and C. Keil, Rigorous error bounds for the optimal value in semidefinite programming. SIAM Journal on Numerical Analysis,46 (2007), 180–200, Scholar
  33. [33]
    R.B. Kearfott, GlobSol. Scholar
  34. [34]
    R.B. Kearfott, On proving existence of feasible points in equality constrained optimization problems. Math. Program.,83 (1998), 89–100.MathSciNetGoogle Scholar
  35. [35]
    R.B. Kearfott, On proving existence of feasible points in equality constrained optimization problems. Preprint, Department of Mathematics, Univ. of Southwestern Louisiana, U.S.L. Box 4-1010, Lafayette, La 70504, 1994.Google Scholar
  36. [36]
    R.B. Kearfott, Rigorous Global Search: Continuous Problems. Kluwer Academic Publisher, Dordrecht, 1996.zbMATHGoogle Scholar
  37. [37]
    C. Keil, Verified linear programming — a comparison. Submitted, 2008, Scholar
  38. [38]
    C. Keil, Lurupa — rigorous error bounds in linear programming. Algebraic and Numerical Algorithms and Computer-Assisted Proofs, B. Buchberger, S. Oishi, M. Plum and S.M. Rump (eds.), Dagstuhl Seminar Proceedings, No. 05391. Internationales Begegnungs- und Forschungszentrum fuer Informatik (IBFI), Schloss Dagstuhl, Germany, 2006, Scholar
  39. [39]
    C. Keil and C. Jansson, Computational experience with rigorous error bounds for the Netlib linear programming library. Reliable Computing,12 (2006), 303–321. Scholar
  40. [40]
    R. Krawczyk, Fehlerabschätzung bei linearer Optimierung, Interval Mathematics, K. Nickel (ed.), Lecture Notes in Computer Science, Vol. 29, Springer-Verlag, Berlin, 1975, 215–222.Google Scholar
  41. [41]
    M. Laurent and S. Poljak, On a positive semidefinite relaxation of the cut polytope. Linear Algebra and Its Applications (LAA),223/224 (1995), 439–461.CrossRefMathSciNetGoogle Scholar
  42. [42]
    J. Löfberg, YALMIP: A toolbox for modeling and optimization in MATLAB. Proceedings of the CACSD Conference, Taipei, Taiwan, 2004.Google Scholar
  43. [43]
    H.D. Mittelmann, An independent benchmarking of SDP and SOCP solvers. Math. Programming Ser. B,95 (2003), 407–430.zbMATHCrossRefMathSciNetGoogle Scholar
  44. [44]
    R.E. Moore, Methods and Applications of Interval Analysis. SIAM, Philadelphia, 1979.zbMATHGoogle Scholar
  45. [45]
    A. Nemirovskii, Lectures on Modern Convex Optimization. 2003.Google Scholar
  46. [46]
    Y. Nesterov, Long-step strategies in interior-point primal-dual methods. Math. Programming,76 (1997), 47–94.MathSciNetGoogle Scholar
  47. [47]
    Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia, 1994.zbMATHGoogle Scholar
  48. [48]
    Y.E. Nesterov and M.J. Todd, Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res.,22 (1997), 1–42.zbMATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    NETLIB Linear Programming Library. Scholar
  50. [50]
    A. Neumaier, Interval Methods for Systems of Equations. Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 1990.Google Scholar
  51. [51]
    A. Neumaier, Introduction to Numerical Analysis. Cambridge University Press, 2001.Google Scholar
  52. [52]
    A. Neumaier, Complete search in continuous global optimization and constraint satisfaction. Acta Numerica, Vol. 13, A. Iserles (eds.), Cambridge University Press, 2004, 271–369.Google Scholar
  53. [53]
    A. Neumaier and O. Shcherbina, Safe bounds in linear and mixed-integer programming. Mathematical Programming, Ser. A,99 (2004), 283–296.zbMATHCrossRefMathSciNetGoogle Scholar
  54. [54]
    J. von Neumann and H.H. Goldstine, Numerical inverting of matrices of high order. Bull. Amer. Math. Soc.,53 (1947), 1021–1099.zbMATHCrossRefMathSciNetGoogle Scholar
  55. [55]
    S. Oishi and S.M. Rump, Fast verification of solutions of matrix equations. Numer. Math.,90 (2002), 755–773.zbMATHCrossRefMathSciNetGoogle Scholar
  56. [56]
    S. Oishi, K. Tanabe, T. Ogita and S.M. Rump, Convergence of Rump’s method for inverting arbitrarily ill-conditioned matrices. J. Comput. Appl. Math.,205 (2007), 533–544.zbMATHCrossRefMathSciNetGoogle Scholar
  57. [57]
    F. Ordóñez and R.M. Freund, Computational experience and the explanatory value of condition measures for linear optimization. SIAM J. Optimization (SIOPT),14 (2003), 307–333.zbMATHCrossRefGoogle Scholar
  58. [58]
    A.L. Peressini, Ordered Topological Vector Spaces. Harper and Row, 1967.Google Scholar
  59. [59]
    J. Renegar, Some perturbation theory for linear programming. Mathematical Programming,65 (1994), 79–91.CrossRefMathSciNetGoogle Scholar
  60. [60]
    J. Renegar, Linear programming, complexity theory, and elementary functional analysis. Mathematical Programming,70 (1995), 279–351, Scholar
  61. [61]
    S.M. Rump, Solving algebraic problems with high accuracy (Habilitationsschrift), A New Approach to Scientific Computation, U.W. Kulisch and W.L. Miranker (eds.), Academic Press, New York, 1983, 51–120.Google Scholar
  62. [62]
    S.M. Rump, Validated solution of large linear systems. Validation Numerics: Theory and Applications, R. Albrecht, G. Alefeld and H.J. Stetter (eds.), Computing Supplementum, Vol. 9, Springer, 1993, 191–212.Google Scholar
  63. [63]
    S.M. Rump, INTLAB—interval laboratory, a Matlab toolbox for verified computations, Version 5.1, 2005.Google Scholar
  64. [64]
    S.M. Rump, Error bounds for extremely ill-conditioned problems. Proceedings of 2006 International Symposium on Nonlinear Theory and Its Applications, Bologna, Italy, September 11–14, 2006.Google Scholar
  65. [65]
    S.M. Rump, INTLAB—interval laboratory, the Matlab toolbox for verified computations, Version 5.3, 2006.Google Scholar
  66. [66]
    S.M. Rump and T. Ogita, Super-fast validated solution of linear systems. Special issue on scientific computing, computer arithmetic, and validated numerics (SCAN 2004), Journal of Computational and Applied Mathematics (JCAM),199 (2006), 199–206.CrossRefMathSciNetGoogle Scholar
  67. [67]
    H.H. Schaefer, Banach lattices and positive operators. Springer, 1974.Google Scholar
  68. [68]
    J.F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software,11 (1999), 625–653.CrossRefMathSciNetGoogle Scholar
  69. [69]
    J.F. Sturm, Central region method. High Performance Optimization, J.B.G. Frenk, C. Roos, T. Terlaky and S. Zhang (eds.), Kluwer Academic Publishers, 2000, 157–194.Google Scholar
  70. [70]
    L. Tuncel, Generalization of primaldual interior-point methods to convex optimization problems in conic form. Found. Comput. Math.,1 (2001), 229–254.zbMATHMathSciNetGoogle Scholar
  71. [71]
    A.M. Turing, Rounding-off errors in matrix processes. Quarterly J. of Mechanics & App. Maths.,1 (1948), 287–308.zbMATHCrossRefMathSciNetGoogle Scholar
  72. [72]
    R.H. Tütüncü, K.C. Toh and M.J. Todd, Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program.,95 (2003), 189–217.zbMATHCrossRefMathSciNetGoogle Scholar
  73. [73]
    P. Van Hentenryck, P. Michel and Y. Deville, Numerica: A Modelling Language for Global Optimization. MIT Press, Cambridge, 1997.Google Scholar
  74. [74]
    H. Wolkowicz, Semidefinite and Cone Programming Bibliography, Comments. Scholar
  75. [75]
    H. Wolkowicz, R. Saigal and L. Vandenberghe (eds.), Handbook of Semidefinite Programming. International Series in Operations Research and Management Science, Vol. 27, Kluwer Academic Publishers, Boston, MA, 2000.Google Scholar
  76. [76]
    M. Yamashita, K. Fujisawa and M. Kojima, Implementation and evaluation of SDPA 6.0. Optimization Methods and Software,18 (2003), 491–505.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© JJIAM Publishing Committee 2009

Authors and Affiliations

  1. 1.Institute for Reliable ComputingHamburg University of TechnologyHamburgGermany

Personalised recommendations