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New Optimization Algorithms for Structural Reliability Analysis

S.R. Santos1, L.C. Matioli2, A.T. Beck3
State University of Paraná, Department of Mathematics, CP 415, 87303-100, Campo Mourão-PR, Brazil. Ph.D. Program in Numerical Methods in Engineering, Federal University of Paraná, Curitiba-PR, Brazil. Partially supported by Fundação Araucária. E-mail:
Federal University of Paraná, Department of Mathematics, Centro Politécnico, CP 19081, 81531-990, Curitiba-PR, Brazil. E-mail:
University of São Paulo, São Carlos School of Engineering, Department of Structural Engineering,13566-590, São Carlos-SP, Brazil. E-mail:

Computer Modeling in Engineering & Sciences 2012, 83(1), 23-56.


Solution of structural reliability problems by the First Order method require optimization algorithms to find the smallest distance between a limit state function and the origin of standard Gaussian space. The Hassofer-Lind-Rackwitz-Fiessler (HLRF) algorithm, developed specifically for this purpose, has been shown to be efficient but not robust, as it fails to converge for a significant number of problems. On the other hand, recent developments in general (augmented Lagrangian) optimization techniques have not been tested in aplication to structural reliability problems. In the present article, three new optimization algorithms for structural reliability analysis are presented. One algorithm is based on the HLRF, but uses a new differentiable merit function with Wolfe conditions to select step length in linear search. It is shown in the article that, under certain assumptions, the proposed algorithm generates a sequence that converges to the local minimizer of the problem. Two new augmented Lagrangian methods are also presented, which use quadratic penalties to solve nonlinear problems with equality constraints. Performance and robustness of the new algorithms is compared to the classic augmented Lagrangian method, to HLRF and to the improved HLRF (iHLRF) algorithms, in the solution of 25 benchmark problems from the literature. The new proposed HLRF algorithm is shown to be more robust than HLRF or iHLRF, and as efficient as the iHLRF algorithm. The two augmented Lagrangian methods proposed herein are shown to be more robust and more efficient than the classical augmented Lagrangian method.


Structural reliability, HLRF-based algorithms, Nonlinear programming, augmented Lagrangian methods.

Cite This Article

Santos, S., Matioli, L., Beck, A. (2012). New Optimization Algorithms for Structural Reliability Analysis. CMES-Computer Modeling in Engineering & Sciences, 83(1), 23–56.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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