Estalishing a nonlinear optimization method for structural reliability analysis

The Hasofer-Lind and Rackwitz-Fiessler (HLRF) algorithm, which is based on the first-order reliability method (FORM), is widely used to estimate failure probability, reliability index, and design point in structural reliability analysis. However, due to the high nonlinearity of the limit state surface, the HLRF algorithm can be unstable. To address this issue, this paper proposes an optimization method to locate and estimate the design point in the standard normal space and calculate the corresponding failure probability. The reliability problem is solved using sequential least squares programming (SLSQP) to improve accuracy, robustness, and efficiency. SLSQP replaces the quadratic programming problem with a linear least-squares problem, using a stable LDL factorization of the Hessian of the Lagrangian equation. The initial optimization problem is converted into a minimum distance optimization problem with a lower bound constraint. To eliminate linearization errors, the probability expectation method with rotation directions space is employed. The proposed algorithm is demonstrated in several benchmark numerical examples with both explicit and implicit limit state functions. Its fast convergence rate is a notable feature of the proposed algorithm, which enhances its competitiveness in structural reliability analysis.