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Structural Reliability Optimization

  • Philippe Geyskens
Conference paper
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)

Abstract

Maximum likelihood points play a predominant role in structural reliability computation. The point of maximum likelihood (or the failure point) is the solution of a nonlinear mathematical programming problem with a nonlinear equality constraint (NEP). Sequential quadratic programming (SQP) is regarded as one of the best procedures to solve an NEP1, 2. The different computational strategies on which the SQP method is based are described. Second-order reliability analysis methods need second-order derivative information of the likelihood and the limit state surface at the optimal point. An optimization-based algorithm for the calculation of the reduced Hessian of the Lagrangian is discussed.

Keywords

Structural reliability nonlinear equality constrained program optimization second-order reliability method 

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References

  1. 1.
    P.E. Gill, W. Murray, and M.H. Wright. Practical Optimization. Academic Press, New York, USA, 1981.MATHGoogle Scholar
  2. 2.
    D.G. Luenberger. Linear and Nonlinear Programming. Addison-Wesley, Reading, Massachusetts, USA, second edition, 1984.MATHGoogle Scholar
  3. 3.
    K. Breitung. Probability approximations by loglikelihood maximization. Journal of Engineering Mechanics, ASCE, 117(3):457–477, 1991.CrossRefGoogle Scholar
  4. 4.
    K. Breitung. Asymptotic approximation for multinormal integrals.Journal of Engineering Mechanics, ASCE, 110(3):357–366, 1984.CrossRefGoogle Scholar
  5. 5.
    M.A. Maes, K. Breitung, and D.J. Dupuis. Asymptotic importance sampling. Structural Safety, 12:167–186, 1993.CrossRefGoogle Scholar
  6. 6.
    Ph. Geyskens. Optimization and Quadrature Algorithms for Structural Reliability Analaysis in the Original Variable Domain. PhD thesis, Department of Civil Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, 1993.Google Scholar
  7. 7.
    M. Shinozuka. Basic analysis of structural safety. Journal of Structural Engineering, ASCE, 109(3):721–740, 1983.CrossRefGoogle Scholar
  8. 8.
    M. Hohenbichler and R. Rackwitz. Non-normal dependent vectors in structural safety. Journal of the Engineering Mechanics Division, ASCE, 107(6):1227–1238, 1981.Google Scholar
  9. 9.
    W. Murray. Linear algebra and optimization. Lecture Notes, Stanford University, Stanford, CA, 1993.Google Scholar
  10. 10.
    A.M. Hasofer and N.C. Lind. Exact and invariant second-moment code format. Journal of the Engineering Mechanics Division, ASCE 100(1):111–121, 1974.Google Scholar
  11. 11.
    R. Rackwitz and B. Fiefiler. Structural reliability under combined random load sequences. Computers & Structures, 9:489–494, 1978.MATHCrossRefGoogle Scholar
  12. 12.
    A. Der Kiureghian and M. De Stefano. Efficient algorithm for second-order reliability analysis. Journal of Engineering Mechanics, ASCE, 117(12):2904–2923, 1991.CrossRefGoogle Scholar
  13. 13.
    P-L. Liu and A. Der Kiureghian. Optimization algorithms for structural reliability analysis. Technical Report UCB/SEMM-86/09, Department of Civil Engineering, University of California at Berkeley, Berkeley, California, USA, 1986.Google Scholar
  14. 14.
    P-L. Liu and A. Der Kiureghian. Optimization algorithms for structural reliability. Structural Safety, 9(3):161–177, 1991.CrossRefGoogle Scholar
  15. 15.
    T. Abdo and R. Rackwitz. A new beta-points algorithm for large time-invariant and time-variant reliability problems. In A. A. Der Kiureghian and P. Thoft-Christensen, editors, Reliability and Optimization of Structural Systems ’90, Proceedings of the 3rd IFIP WG7.5 Conference, Berkeley, CA, pages 1–11, Berlin, Germany, 1990. Springer-Verlag.Google Scholar
  16. 16.
    P.E. Gill and W. Murray. Numerically stable methods for quadratic programming. Mathematical Programming, 14:349–372, 1978.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    G.H. Golub and C.F. Van Loan. Matrix Computations. The John Hopkins University Press, Baltimore, Maryland, USA, second edition, 1989.MATHGoogle Scholar
  18. 18.
    J.E. Dennis and J.J. Moré. Quasi-newton methods, motivation and theory. SIAM Review, 19(l):46–89, 1977.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    M.J.D. Powell. The Convergence of Variable Metric Methods for Nonlinearly Constrained Optimization Problems, pages 27–63. Academic Press, New York, USA, 1978.Google Scholar
  20. 20.
    L.C.W. Dixon. The Choice of Step Length, a Crucial Factor in the Performance of Variable Metric Algorithms, chapter 10, pages 149–170. Academic Press, London, UK, 1972.Google Scholar
  21. 21.
    K. Schittkowski. The nonlinear programming method of wilson, han and powell with an augmented lagrangian type line search function - part i: Convergence analysis. Numerische Mathematic, 38:83–114, 1981.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    K. Schittkowski. On the convergence of a sequential quadratic programming method with an augmented lagrangian line search function. Mathematische Operationsforschung und Statistic, series Optimization, 14(2): 197–216, 1983.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    R.P. Brent. Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs, New Jersey, USA, 1973.MATHGoogle Scholar
  24. 24.
    D.P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, USA, 1982.MATHGoogle Scholar
  25. 25.
    P.T. Boggs, J.W. Tolle, and P. Wang. On the local convergence of quasi-new ton methods for constrained optimization. SIAM Journal of Control and Optimization, 20(2):161–171, 1982.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    P.T. Boggs and J.W. Tolle. Convergence propoerties of a class of rank-two updates. Technical Report UNC/OR TR-90/16, Department of Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, USA, 1992.Google Scholar
  27. 27.
    M.R. Hestenes. Conjugate Directions Methods in Optimization. Springer-Verlag, New York, USA, 1980.Google Scholar
  28. 28.
    T. Igusa and A. Der Kiureghian. Dynamic characterization of two-degree-of-freedom equipment-structure systems. Journal of Engineering Mechanics, ASCE, 111(1):1–19, 1985.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag, Berlin Heidelberg 1994

Authors and Affiliations

  • Philippe Geyskens
    • 1
  1. 1.Department of Civil EngineeringKatholieke Universiteit LeuvenBelgium

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