## 3 Optimization Scheme
### The Label-Clip-Slice Strategy and GA to Address Mixed Variables
#### 3.3.4 Splicing of the Individuals in the Current Population  
In the iterative process, the variables labeled \(R_{bit}\) are spliced to their original positions for the above population before checking the validity of the structure configuration and calculating the individuals' fitness. In the two stages, the topology variables will not be operated on but the size variables will be. The shaded bits are the label variables, and they are spliced to each individual, as shown in Figure 4. This is because the validity check needs the individual's physical meaning, which consists of all the initial variables. In addition, the labeled variables have almost no effect on the fitness calculation of GA.  
Figure 3: Clipping of the chromosome of the topology variable.  
\(j\)-th constraint obtained by the first-order [PERSON] expansion of reciprocal design variables in the \(m\)-th step, \(\tilde{x}_{i(m)}^{ll}\) and \(\tilde{x}_{i(m)}^{l}\) are move limits of \(\tilde{x}_{i}\) in the \(m\)-th step. \(\overline{x}_{i(m)}^{ll}\) and \(\overline{x}_{i(m)}^{l}\) are the upper and lower bounds of variables \(\tilde{x}_{i}\). After the construction of the second-level approximation problem, a dual method is implemented to transform the constraint optimization problem into an unconstrained problem. Then, an unconstrained optimization algorithm BFGS is used to seek the optimization size variable \(\mathbf{X}_{l,k,p}^{*}\).  
After fitness calculation and obtainment of the optimal size results, the labeled bits for all individuals are clipped again before conducting the genetic roulette wheel selection, crossover, and mutation operator procedures in sequence due to their only processing topology variables. When the maximum generation is reached in every run of the GA, it goes to the elite selection strategy. A few (i.e., 5) appropriate individuals (the individual with the highest fitness value and four individuals of the lowest weight) are selected. Among them, the individual \(\mathbf{X}_{p,clip}^{*}\) that satisfies the constraints will be selected as the elite individual and reserved for the next generation.  
\[\left\{\begin{array}{l}\left|x_{ip}-x_{i(p-1)}\right|\leq\varepsilon_{x}\;(i =1,2,\cdots,n)\\ \\ \mathrm{Max}\Big{(}g_{1}^{(p)}\big{(}\mathbf{X}_{p},\mathbf{\alpha}_{p}\big{)},g_{2}^ {(p)}\big{(}\mathbf{X}_{p},\mathbf{\alpha}_{p}\big{)},\cdots,g_{l}^{(p)}\big{(}\mathbf{X}_ {p},\mathbf{\alpha}_{p}\big{)}\Big{)}\leq\varepsilon_{g}\;\text{or}\;p\leq p_{ \max}\\ \\ \left|\frac{W^{(p)}\big{(}\mathbf{X}_{p},\mathbf{\alpha}_{p}\big{)}-W^{(p)}\big{(}\mathbf{X }_{p-1},\mathbf{\alpha}_{p-1}\big{)}}{W^{(p)}\big{(}\mathbf{X}_{p},\mathbf{\alpha}_{p} \big{)}}\right|\leq\varepsilon_{f}\end{array}\right. \tag{16}\]  
The convergence criterion defined in Equation (16) is used to determine whether the first-level approximation problem is terminated, as well as the whole scheme. \(\varepsilon_{x}\) is the convergence control parameter of size variables, \(\varepsilon_{g}\) is the constraints control parameter, \(\varepsilon_{f}\) is the weight convergence control parameter, and \(p_{\max}\) is the maximum iterative number for the first-level approximation problem. In this paper, \(\varepsilon_{x}\)= 0.0001, \(\varepsilon_{g}\) = 0.05, \(\varepsilon_{f}\) = 0.001, and \(p_{\max}\) = 100; the maximum number of second-level iterations for sizing optimization is 20 with a convergence accuracy of 0.001.