Mathematical background and formulation of numerical minimization process are described in terms of gradient-based methods, whose ingredients include gradient, Hessian, directional derivatives, optimality conditions for minimization, Hessian eigensystem, conjugate number of Hessian, and conjugate vectors. Various minimization algorithms, such as the steepest descent method, Newton’s method, conjugate gradient method, and quasi-Newton’s method, are introduced along with practical examples.

A brief derivation of the Taylor expansion for a scalar function depending on a vector, with application to the inverse distance dependence in the Coulomb expression for electrostatic interactions.

Chapter 7 describes the second-order reliability method (SORM), which employs a second-order approximation of the limit-state surface fitted at the design point in the standard normal space. Three distinct SORM approximations are presented. The classical SORM fits the second-order approximating surface to the principal curvatures of the limit-state surface at the design point. This approach requires computing the Hessian (second-derivative matrix) of the limit-state function at the design point and its eigenvalues as the principal curvatures. The second approach computes the principal curvatures iteratively in the process of finding the design point. This approach requires only first-order derivatives of the limit-state function but repeated solutions of the optimization problem for finding the design point. One advantage is that the principal curvatures are found in decreasing order of magnitude and, hence, the computations can be stopped when the curvature found is sufficiently small. The third approach fits the approximating second-order surface to fitting points in the neighborhood of the design point. This approach also avoids computing the Hessian. Furthermore, it corrects for situations where the curvature is zero but the surface is curved, e.g., when the design point is an inflection point of the surface. Results from the three methods are compared numerically.

A brief review of optimization techniques used in the book.