Issue 78

A. Gallo et alii, Fracture and Structural Integrity, 78 (2026) 489-503; DOI: 10.3221/IGF-ESIS.78.28

Comparisons with first-order, second- order, and Levenberg–Marquardt optimization methods were also performed to evaluate convergence reliability and computational efficiency. The results provided insight into the behaviour of the algorithm across different optimization applications.

M ATERIALS AND METHODS O PTIMIZATION PROBLEMS AND N EWTON ALGORITHM

Many engineering problems can be formulated as constrained or unconstrained optimization problems, with the aim of identifying the set of design variables that minimizes a scalar objective function. Let ሺ ሻ be a scalar function defined on ℝ ௡ , where is the vector of design variables, a general iterative scheme, named line search, for minimizing ሺ ሻ can be expressed as [7,14,18]: ௞ାଵ ൌ ௞ ൅ ∙ ௞ (1) where ௞ is the vector of design variables at iteration k , the positive value is the step-size, and ௞ is the search direction. To ensure that ௞ is effectively a descent direction, the following condition must hold: ∇ ൫ ௞ ൯் ∙ ௞ ൏ 0 (2) assuming sufficiently small and ∈ ଶ ሺℝ ௡ ሻ . The search direction can be generally defined as: ௞ ൌെ ି ଵ ൫ ௞ ൯ ∙ ∇ ൫ ௞ ൯ (3) where A is a non-singular symmetric second order tensor. Different optimization algorithms are obtained depending on the definition of the tensor A. The Gradient Descent method, i.e. a classic first-order method, when ൌ , where I is the identity tensor. The Newton method, i.e. a second order method, when ൌ , where H is the Hessian tensor assumed to be symmetric. The Quasi-Newton method when ≅ , where A is an approximation of the Hessian tensor at the current iteration. The Newton method can be derived from the second-order Taylor expansion of the objective function around the current iteration: ሺ ሻ ≅ ൫ ௞ ൯൅∇ ൫ ௞ ൯் ∙ ൅ ଵ ଶ ் ∙ ൫ ௞ ൯ ∙ (4) with ൌ ௞ାଵ െ ௞ (5) From the stationary condition given by ሺ ሻ െ ൫ ௞ ൯≅ 0 and with simple calculation steps, the minimization of the quadratic approximation leads to the standard Newton update: ൌെ ି ଵ ൫ ௞ ൯ ∙ ∇ ൫ ௞ ൯ (6) By substituting eq. (6) into eq. (1), the Newton iteration formula can be written as ௞ାଵ ൌ ௞ െ ∙ ି ଵ ൫ ௞ ൯ ∙ ∇ ൫ ௞ ൯ (7) The classic Newton method is obtained for step-size equal to unity ሺ ൌ 1 ሻ , whilst for ് 1 the method is commonly referred to as modified or relaxed Newton. Although the standard Newton algorithm converges in a single iteration for quadratic objective functions and exhibits quadratic convergence for sufficiently smooth functions [7,14,18], it may diverge for highly nonlinear objective functions

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