Issue 78

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

or poor initial guesses. For this reason, practical implementations typically combine the Newton direction with line search strategies.

C ONSTRAINED OPTIMIZATION USING RELAXED N EWTON METHOD COMBINED WITH BACKTRACKING TECHNIQUE et ∈ ଶ ሺℝሻ be a scalar objective function defined on the design variable vector ∈ℝ ௡ , such that : ℝ ௡ → ℝ . In this work, the number of design variables ranges between 2 and 9, depending on the considered application, whilst the objective function ሺ ሻ can be either numerical or analytical-numerical. The search direction is computed by Newton formulation using the Hessian tensor in eq. (3), whilst the step size is determined through a backtracking line-search procedure. Since the location of the minimum is generally unknown, each iteration must effectively ensure a decrease of the objective function [7,11,14,18]. To meet this requirement, the step size can be selected according to the Wolfe conditions. The first Wolfe condition imposes a sufficient reduction of the objective function: ൫ ௞ାଵ ൯ൌ ൫ ௞ ൅ ∙ ௞ ൯൑ ൫ ௞ ൯ ൅ ∙ ∙ ∇ ൫ ௞ ൯் ∙ ௞ (8) where ∈ ሺ 0,1 ሻ is a control parameter. The higher the value of , the more restrictive the Armijo rule. The second Wolfe condition, instead, requires: ∇ ൫ ௞ାଵ ൯் ∙ ௞ ൒ ∙ ∇ ൫ ௞ ൯் ∙ ௞ (9) where ∈ ሺ ,1 ሻ is a control parameter of the curvature condition. This condition suggests that a strongly negative directional derivative slope identifies a direction of significant decrease in the objective function [19]. Together, these conditions specify the allowable range for the step size: ൝ ൫ ௞ ൅ ∙ ௞ ൯൑ ൫ ௞ ൯ ൅ ∙ ∙ ∇ ൫ ௞ ൯் ∙ ௞ ∇ ൫ ௞ାଵ ൯் ∙ ௞ ൒ ∙ ∇ ൫ ௞ ൯் ∙ ௞ (10) For highly nonlinear functions, the satisfaction of both Wolfe conditions may become computationally expensive. Therefore, in the present study a modified backtracking strategy is adopted, in which only the Armijo condition is explicitly evaluated [19], while the curvature condition is indirectly enforced through a continuous step-size refinement factor , which is defined as ൌ 2ି ௝ , ൌ ̅൅ (11) where ̅ ൌ 0,1,…, with ∈ℕ and ∈ ሺ 0,1 ሿ . This formulation progressively reduces the trial step length until the sufficient decrease condition is satisfied. Termination criterion of optimization process is based on both the reaching of maximum number of iterations and the stabilization of the objective function value. Convergence is assumed when three consecutive iterations produce negligible variations in the function ሺ ሻ value. For constrained optimization problems, equality and inequality constraints are additionally verified throughout the numerical procedure to ensure feasibility. In standard backtracking algorithms, as is well known in the literature, the admissible step lengths are restricted to the discrete sequence ൌ 2ି ௝ with ∈ℕ 0 , which generates a finite set of suitable step lengths. Although this choice is computationally efficient in many engineering applications, its discretization may become excessively coarse in highly nonlinear finite element optimization problems, where the objective function exhibits strong non- convexity and multiple local minima. To overcome this limitation, an extension of the mathematical structure of the standard backtracking line-search strategy is developed by means of the continuous relaxation parameter , obtaining a finer discretization of the search space without modifying the Newton search direction, introducing additional optimization variables, or requiring auxiliary line-search algorithms. Unlike conventional tuning parameters, does not depend on the physical problem, constitutive model or objective function under consideration. Depending on the adopted solution strategy, can be defined at the start of the L

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