Advertisement

A Review of Techniques in the Verified Solution of Constrained Global Optimization Problems

  • R. Baker Kearfott
Part of the Applied Optimization book series (APOP, volume 3)

Abstract

Elements and techniques of state-of-the-art automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previously developed algorithms and general work on the subject are also listed. Limitations of present knowledge are mentioned, and advice is given on which techniques to use in various contexts. Applications are discussed.

Keywords

Global Optimization Interval Arithmetic Global Optimization Algorithm Interval Vector Uniqueness Verification 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York, 1983.zbMATHGoogle Scholar
  2. [2]
    E. Baumann, “Optimal Centered Forms”, BIT, 1988, Vol. 28, No. 1, pp. 80–87.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    O. Caprani, B. Godthaab, and K. Madsen, “Use of a Real-Valued Local Minimum in Parallel Interval Global Optimization”, Interval Computations, 1993, No. 2, pp. 71–82.MathSciNetGoogle Scholar
  4. [4]
    A. R. Conn, N. Gould, and Ph. L. Toint, LANCELOT: A Fortran Package for Large-Scale Nonlinear Optimization, Springer-Verlag, New York, 1992.zbMATHGoogle Scholar
  5. [5]
    T. Csendes, “Nonlinear Parameter Estimation by Global Optimization — Efficiency and Reliability”, Acta Cybernetica, 1988, Vol. 8, No. 4, pp. 361–370.zbMATHMathSciNetGoogle Scholar
  6. [6]
    T. Csendes and D. Ratz, Subdivision Direction Selection in Interval Methods for Global Optimization, preprint, 1994.Google Scholar
  7. [7]
    J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Least Squares, Prentice-Hall, Englewood Cliffs, NJ, 1983.Google Scholar
  8. [8]
    Kaisheng Du and R. B. Kearfott, “The Cluster Problem in Global Optimization The Univariate Case”, Computing (Suppl.), 1992, Vol. 9, pp. 117–127.Google Scholar
  9. [9]
    W. Enger, “Interval Ray Tracing — A Divide and Conquer Strategy for Realistic Computer Graphics”, The Visual Computer, 1992, Vol. 9, pp. 91–104.CrossRefGoogle Scholar
  10. [10]
    J. Eriksson, Parallel Global Optimization using Interval Analysis, Ph.D. dissertation, University of Umeå, Institute of Information Processing, 1991.Google Scholar
  11. [11]
    J. Eriksson and P. Lindstroem, “A Parallel Interval Method Implementation for Global Optimization Using Dynamic Load Balancing”, Reliable Computing, Vol. 1, No. 1, pp. 77–92.Google Scholar
  12. [12]
    C. A. Floudas and P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms, Springer-Verlag, New York, 1990.zbMATHGoogle Scholar
  13. [13]
    P. E. Gill, W. Murray, and M. Wright, Practical Optimization, Academic Press, New York, 1981.zbMATHGoogle Scholar
  14. [14]
    G. D. Hager, Solving Large Systems of Nonlinear Constraints with Application to Data Modeling, preprint, 1993.Google Scholar
  15. [15]
    R. Hammer, M. Hocks, U. Kulisch, and D. Ratz, Numerical Toolbox for Verified Computing I, Springer-Verlag, New York, 1993.zbMATHGoogle Scholar
  16. [16]
    E. R. Hansen, “Interval Forms of Newton’s Method”, Computing, 1978, Vol. 20, pp. 153–163.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    E. R. Hansen, “Global Optimization Using Interval Analysis: The One- Dimensional Case”, J. Optim. Theory Appl, 1979, Vol. 29, No. 3, pp. 331–344.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    E. R. Hansen, “Global Optimization Using Interval Analysis: the Multidimensional Case”, Numer. Math., 1980, Vol. 34, No. 3, pp. 247–270.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    E. R. Hansen, Global Optimization Using Interval Analysis, Marcel Dekker, Inc., New York, 1992.zbMATHGoogle Scholar
  20. [20]
    E. R. Hansen and G. W. Walster, “Bounds for Lagrange Multipliers and Optimal Points”, Comput. Math. Appl, 1993, Vol. 25, No. 10, pp. 59–ff.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    K. Ichida and Y. Fujii, “An Interval Arithmetic Method for Global Optimization”, Computing, 1979, Vol. 23, No. 1, pp. 85–97.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    C. Jansson, A Global Minimization Method: The One-Dimensional Case, Technical Report No. 91. 2, 1991.Google Scholar
  23. [23]
    C. Jansson, “A Global Optimization Method Using Interval Arithmetic”, In: L. Atanassova and J. Herzberger (eds.), Computer Arithmetic and Enclosure Methods. Proc. Third International IMACS-GAMM Symposium on Computer Arithmetic and Scientific Computing, North-Holland, Amsterdam, Netherlands, 1992, pp. 259–268.Google Scholar
  24. [24]
    C. Jansson and O. Knüppel, A Global Minimization Method: The Multi- Dimensional Case, preprint, 1992.Google Scholar
  25. [25]
    C. Jansson, “On Self-Validating Methods for Optimization Problems”, In: J. Herzberger (ed.), Topics in Validated Computations, North-Holland, Amsterdam, Netherlands, 1994, pp. 381–439.Google Scholar
  26. [26]
    C. Jansson and O. Knüppel, Numerical Results for a Self-Validating Global Optimization Method, Technical Report, 1994.Google Scholar
  27. [27]
    R. B. Kearfott, “Abstract Generalized Bisection and a Cost Bound”, Math. Comp., 1987, Vol. 49, No. 179, pp. 187–202.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    R. B. Kearfott, “Interval Arithmetic Techniques in the Computational Solution of Nonlinear Systems of Equations: Introduction, Examples, and Comparisons”, In: E. L. Allgower and K. Georg (eds.), Computational Solution of Nonlinear Systems of Equations (Lectures in Applied Mathematics, Vol. 26), American Mathematical Society, Providence, RI, 1990, pp. 337–358.Google Scholar
  29. [29]
    R. B. Kearfott, “Preconditioned for the Interval Gauss-Seidel Method”, SIAM J. Numer. Anal., 1990, Vol. 27, No. 3, pp. 804–822.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    R. B. Kearfott and M. Novoa, “INTBIS, A Portable Interval Newton/Bisection Package (Algorithm 681)”, ACM Trans. Math. Software, 1990, Vol. 16, No. 2, pp. 152–157.zbMATHCrossRefGoogle Scholar
  31. [31]
    R. B. Kearfott, C. Y. Hu, and M. Novoa III, “A Review of Preconditioners for the Interval Gauss-Seidel Method”, Interval Computations, 1991, No. 1, pp. 59–85.MathSciNetGoogle Scholar
  32. [32]
    R. B. Kearfott, “An Interval Branch and Bound Algorithm for Bound Constrained Optimization Problems”, Journal of Global Optimization, 1992, Vol. 2, pp. 259–280.zbMATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    R. B. Kearfott, M. Dawande, K.-S. Du, and C.-Y. Hu, “Algorithm 737: INTLIB: A Portable FORTRAN 77 Interval Standard Function Library” ACM Trans. Math. Software, 1994, Vol. 20, No. 4, pp. 447–459.zbMATHCrossRefGoogle Scholar
  34. [34]
    R. B. Kearfott, “A Fortran 90 Environment for Research and Prototyping of Enclosure Algorithms for Constrained and Unconstrained Nonlinear Equations”, ACM Trans. Math. Software, 1995, Vol. 21, No. 1, pp. 63–78.zbMATHCrossRefGoogle Scholar
  35. [35]
    R. B. Kearfott, Empirical Evaluation of Innovations in Interval Branch and Bound Algorithms for Nonlinear Algebraic Systems, preprint, 1994.Google Scholar
  36. [36]
    R. B. Kearfott, On Verifying Feasibility in Equality Constrained Optimization Problems, preprint, 1994.Google Scholar
  37. [37]
    R. B. Kearfott and K. Du, “The Cluster Problem in Multivariate Global Optimization”, Journal of Global Optimization, 1994, Vol. 5, pp. 253–265.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    R. Krawczyk, “Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlershranken”, Computing, 1969, Vol. 4, pp. 187–201.zbMATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    B. P. Kristinsdottir, Z. B. Zabinsky, T. Csendes, M. E. Tuttle, “Methodologies for Tolerance Intervals”, Interval Computations, 1993, No. 3, pp. 133–147.MathSciNetGoogle Scholar
  40. [40]
    A. Leclerc, “Parallel Interval Global Optimization in C++”, Interval Computations, 1993, No. 3, pp. 148–163.MathSciNetGoogle Scholar
  41. [41]
    G. Mayer, Epsilon-Inflation in Verification Algorithms, preprint, 1993.Google Scholar
  42. [42]
    R. E. Moore, “A Test for Existence of Solutions to Nonlinear Systems”, SIAM J. Numer. Anal., 1977, Vol. 14, No. 4, pp. 611–615.zbMATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    R. E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.zbMATHGoogle Scholar
  44. [44]
    R. E. Moore and H. Ratschek, “Inclusion Functions and Global Optimization II”, Math. Prog., 1988, Vol. 41, No. 3, pp. 341–356.zbMATHCrossRefMathSciNetGoogle Scholar
  45. [45]
    R. E. Moore, E. Hansen, and A. Leclerc, “Rigorous Methods for Parallel Global Optimization”, In: A. Floudas and P. Pardalos (eds.), Recent Advances in Global Optimization Princeton Univ. Press, Princeton, N.J., 1992, pp. 321–342.Google Scholar
  46. [46]
    J. J. Moré and S. J. Wright, Optimization Software Guide, SIAM, Philadelphia, 1993.zbMATHGoogle Scholar
  47. [47]
    S. P. Mudur and P. A. Koparkar, “Interval Methods for Processing Geometric Objects”, IEEE Comput. Graphics and Appl, 1984, Vol. 4, No. 2, pp. 7–17.CrossRefGoogle Scholar
  48. [48]
    A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, England, 1990.zbMATHGoogle Scholar
  49. [49]
    A. Neumaier, Second-Order Sufficient Optimality Conditions for Local and Global Nonlinear Programming, preprint, 1994.Google Scholar
  50. [50]
    P. M. Pardalos and J. B. Rosen, Constrained Global Optimization: Algorithms and Applications, Springer-Verlag, New York, 1987.zbMATHCrossRefGoogle Scholar
  51. [51]
    P. M. Pardalos and S. A. Vavasis, “Quadratic Programming with One Negative Eigenvalue is NP-Hard”, Journal of Global Optimization, 1992, Vol. 1, No. 1.Google Scholar
  52. [52]
    H. Ratschek and J. Rokne, New Computer Methods for Global Optimization, Wiley, New York, 1988.zbMATHGoogle Scholar
  53. [53]
    H. Ratschek and R. L. Voller, “Global Optimization over Unbounded Domains”, SIAM J. Control Optim 1990, Vol. 28, No. 3, pp. 528–539.zbMATHCrossRefMathSciNetGoogle Scholar
  54. [54]
    D. Ratz, Automatische Ergebnisverifikation bei globalen Optimierungsproblemen, Ph.D. dissertation, Universität Karlsruhe, 1992.zbMATHGoogle Scholar
  55. [55]
    D. Ratz, “Box-Splitting Strategies for the Interval Gauss-Seidel Step in a Global Optimization Method”, Computing, 1994, Vol. 53, pp. 337–354.zbMATHCrossRefMathSciNetGoogle Scholar
  56. [56]
    J. Röhn, NP-Hardness Results for Linear Algebraic Problems with Interval Data, preprint, 1994.Google Scholar
  57. [57]
    S. M. Rump, Kleine Fehlerschranken bei Matrixproblemen, Ph.D. dissertation, Universität Karlsruhe, 1980.zbMATHGoogle Scholar
  58. [58]
    S. M. Rump, “Verification Methods for Dense and Sparse Systems of Equations”, In: J. Herzberger (ed.), Topics in Validated Computations, North- Holland, Amsterdam, 1994, pp. 63–135.Google Scholar
  59. [59]
    C. A. Schnepper, Large Grained Parallelism in Equation-Based Flowsheeting Using Interval Newton/Generalized Bisection Techniques, Ph.D. dissertation, University of Illinois, Urbana, Department of Chemical Engineering, 1992.Google Scholar
  60. [60]
    C. A. Schnepper and M. A. Stadtherr, “Application of a Parallel Interval Newton/Generalized Bisection Algorithm to Equation-Based Chemical Process Flowsheeting”, Interval Computations, 1993, No. 4, pp. 40–64.MathSciNetGoogle Scholar
  61. [61]
    T. W. Sederberg and S. R. Parry, “Comparison of Three Curve Intersection Algorithms”, Comput. Aided Des., 1986, Vol. 18, No. 1, pp. 58–63.CrossRefGoogle Scholar
  62. [62]
    E. C. Sherbrooke and N. M. Patrikalakis, Computation of the Solutions of Nonlinear Polynomial Systems, preprint, 1993.Google Scholar
  63. [63]
    X. Shi and R. B. Kearfott, Some Results on the Regularity of an Interval Matrix, preprint, 1994.Google Scholar
  64. [64]
    X. Shi, Intermediate Expression Preconditioning and Verification for Rigorous Solution of Nonlinear Problems, Ph.D. dissertation, University of Southwestern Louisiana, Department of Mathematics, August 1995.Google Scholar
  65. [65]
    S. Skelboe, “Computation of Rational Interval Functions”, BIT, 1974, Vol. 14, pp. 87–95.zbMATHCrossRefMathSciNetGoogle Scholar
  66. [66]
    G. W. Walster, E. R. Hansen, and S. Sengupta, “Test Results for a Global Optimization Algorithm”, In: P. T. Boggs, R. H. Byrd, and R. B. Schnabel (eds.), Numerical Optimization 1984, SIAM, Philadelphia, 1985, pp. 272–287.Google Scholar
  67. [67]
    M. A. Wolfe, “An Interval Algorithm for Constrained Global Optimization”, J. Comput. Appl. Math., 1994, Vol. 50, pp. 605–612.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • R. Baker Kearfott
    • 1
  1. 1.Department of MathematicsUniversity of Southwestern LouisianaLafayetteUSA

Personalised recommendations