## 3 Numerical Examples
### Pareto Optimal Solutions for a Mass-Spring-Damper System  
In the first example, the aim was to illustrate the utility of the NP constraint in finding the optimum solution. Considering the three objective function terms, the multi-objective optimization problem was formulated as follows:  
\[\min_{m,\ c,\ k}F=\sum_{i=1}^{n}\{\alpha|G(\omega_{i})|+\beta|G^{\prime}(\omega_ {i})|+\gamma(1/\omega_{n})\} \tag{9}\]  
\[Subject\ to\qquad\left\{ \begin{aligned} &-\Lambda_{ij}+\varepsilon\leq 0\\ &\mathbf{h}_{lb}\leq\mathbf{h}_{nb}\leq\mathbf{h}_{ub}\end{aligned} \right. \tag{10}\]  
where the weightings were defined as \(0\leq\alpha\leq 1\), \(0\leq\beta\leq 1\), and \(\alpha+\beta+\gamma=1\). \(\Lambda_{ij}\) denotes the Pick matrix given in Equation (1), and its dimension is determined by the number of discrete frequency values, \(\omega_{i}\). \(\varepsilon\) is a very small value (\(\varepsilon\ll 1\)) (here, \(\varepsilon=10^{-3}\)) used to eliminate spurious oscillations in the positive definiteness check, and \(\mathbf{h}\) is the vector of the structural parameters, i.e., \(\mathbf{h}=\{m\ c\ k\}\), in this case bounded by the lower and upper limit values \(\mathbf{h}_{lb}\) and \(\mathbf{h}_{ub}\), respectively. The input parameters are listed in Table 1, where \(\mathbf{h}_{initial}\) is the initial parameter vector, \(\Delta\omega\) is the frequency increment used in the frequency domain calculations, \(\omega_{vec1}\) and \(\omega_{vec2}\) are frequency vectors used for calculation when \(|G(\omega_{i})|\), and \(|G^{\prime}(\omega_{i})|\), and \(\omega_{d}\) is the damped natural frequency. The lower bound vector \(\mathbf{h}_{lb}\) and the upper bound vector \(\mathbf{h}_{ub}\) were selected such that the mass is in a physically realizable range for experimental setups, the damping coefficient was in the range for metal materials and the stiffness coefficient was in a practical range. The parameter vector range was kept intentionally large to make the differences between the solutions with and without the NP constraint more clear. The interior point method in Matlab was utilized in this example. Note that the terms in Equation (9) are the magnitude of the frequency response \(|G(\omega_{i})|\)  
Figure 3: Flowchart of the optimization algorithm under the NP constraint. the roll rate of the frequency response \(|G^{\prime}(\omega_{i})|\), and the reciprocal of the natural frequency \(1/\omega_{n}\). Due to Equations (7) and (8), the minimization of \(|G(\omega_{i})|\) and \(|G^{\prime}(\omega_{i})|\) provides, respectively, lower displacement and slope \(|G^{\prime}(\omega_{i})|\) around natural frequencies, while the minimization of \(1/\omega_{n}\) provides the maximization of the natural frequency. In short, the minimization of \(|G(\omega_{i})|\) and \(|G^{\prime}(\omega_{i})|\) is equivalent to the minimization of compliance and its rate in the frequency domain. Figure 4 illustrates the Pareto optimal solution sets obtained by increasing the values of \(\alpha\), \(\beta\), and \(\gamma\) by a step size of 0.1, both with and without the NP constraint equations. It can be observed in Figure 4 that the solutions obtained without the NP constraint show more scattering. In the end, using the NP constraints resulted in a total cost function of 108.11, while without using the NP constraints the total cost function was 118.6. Table 2 also shows the optimum solutions calculated by altering the weight of each objective function. We observed that introducing a greater penalty for an objective resulted in lower values for that objective. Table 2 compares the results of three cost functions, which differ from each other due to the scaling factor of each objective: \(F_{1}=\sum_{i=1}^{n}\{\alpha|G(\omega_{i})|+\beta|G^{\prime}(\omega_{i})|+ \gamma(1/\omega_{n})\}\), \(F_{2}=\sum_{i=1}^{n}\{10\alpha|G(\omega_{i})|+\beta|G^{\prime}(\omega_{i})|+ \gamma(1/\omega_{n}\}\), and \(F_{3}=\sum_{i=1}^{n}\{\alpha|G(\omega_{i})|+10\beta|G^{\prime}(\omega_{i})|+ \gamma(1/\omega_{n}\}\).