## 1 Introduction
In this study, the weighting method, which is an a posteriori method, was employed to solve multi-objective optimization problems. Furthermore, some specific objective functions were used to optimize structural components, such as the integral of the weighted relative frequency response [5], the frequency response deviation [32], the weighted sum of the difference between the current and initial thrust coefficients and the torque coefficient [10], and the simultaneous minimization of stress and stiffness for topology optimization [33]. Although there are various objective functions involved in the structural optimization problems mentioned so far, the ultimate goal, or at least a constraint that cannot be ignored, in designing and optimizing a mechanical system is its durability. For this purpose, fatigue life has been considered a constraint [34; 35; 36; 37; 38; 39; 40; 41] or an objective function in different forms in the time domain optimization of structures in various studies. These expressions have included damage [42], a specific function expression [43], multiple functions derived from the statistical properties of fatigue life (standard deviation and mean) [44], and impact damage area [27]. Conversely, it is noteworthy that only several studies considering fatigue life in frequency domain optimization [12; 16; 45] exist in the literature. In this respect, this paper contributes to frequency domain optimization research with numerical simulations defining objective functions, including the fatigue damage to a structure. Frequency domain optimization approaches use both traditional and modern techniques. For instance, nonlinear [2] and linear programming algorithms [4] and methods of moving asymptotes [9; 13; 46] exemplify the traditional optimization techniques used for structural optimization. In addition, modern optimization techniques, such as genetic (GA) and evolutionary algorithms (EA) [6; 7; 16; 18], particle swarm optimization (PSO) [8], harmony search [14], and colliding bodies optimization [47], are also used to optimize the frequency domain. In addition to traditional and modern optimization methods, a combination of both procedures, namely a hybrid approach, may also be deployed to achieve more efficient and accurate calculations. The optimum solutions obtained in the frequency domain exhibit difficulties when applied to physical systems. There remains a need to verify the accuracy of frequency domain results via their comparison with time domain results. Despite the aforementioned advantages of frequency domain analysis, it was demonstrated in [16] that frequency domain analysis may yield infeasible results. In short, while a number of studies exist on structural optimization in the frequency domain, trade-off equations that limit the achievable best solution have not been investigated in the physical domain. Rectifying this issue constitutes the focus of this paper and is achieved using the Nevanlinna-Pick (NP) interpolation theory [48]. Although NP interpolation problems have been studied in control theory [49; 50], they have not been investigated for structural optimization problems. Motivated by this fact, NP constraints are implemented in frequency domain optimization formulations to develop an approach to solving constrained optimization problems by considering reachability constraints. The principal contribution of this study is that the NP equations are imposed as nonlinear constraints, frequency response amplitudes are calculated during optimization iterations, and the numerical results are compared to those obtained without the NP constraint. The design variables that cannot satisfy the NP constraints are eliminated, and the feasible region in the parameter space is used in the search for optimal solutions.