## 3 Optimization Scheme
### Analysis and Discussion  
The label-clip-splice strategy is proposed to solve the problem of the large search domain of GA resulting in the bad approximation effect of BMA for large-scale truss optimization. Its efficacy is analyzed and discussed from the understanding of the problem without the LCS strategy.  
The number of whole topology variables is set as \(n\), and the topology variable \(\mathbf{\alpha}=\{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\}\); then, the total search domain of topology variables in the current iteration for the GA is \(\Omega=2^{n}\). Here, the iteration means the solution process of the first-level approximation problem. \(m\) is denoted as the maximum number of topology variables changed with respect to the last iteration's optimal result after a GA process. Obviously, \(m\) increases when \(n\) becomes bigger. This means that the two optimal topology variables between two consecutive iterations are very different for large-scale problems. In other words, the new updated points will be far away from the known points that construct the first-level approximation problem. The accuracy of approximation will decrease with more iterations. As a result, the GA will search with very inaccurate functions, wander locally in a huge domain and lose the right direction for optimization. A more serious result is that the iteration will diverge. The former result is illustrated schematically in Figure 5. The global optimum \(x^{*}\) is marked by a red point. The purple area indicates a distribution domain \(\Omega_{i}\) of some optimum individuals in the GA of each iteration, and the yellow area indicates a total search domain \(\Omega\). Owing to the inaccuracy of approximation caused by large-scale variables of GA, the optimization scheme without the LCS strategy is almost impossible to catch \(x^{*}\) in \(\Omega_{i}\) (\(i\) = 1, 2,..., \(n\)), as shown in the Figure 5.  
The LCS technique shrinks the search domain by decreasing the dimensions of topology variables. The optimal result is saved in each iteration. In the \(p\)-th iteration of the first-level approximation problem, the last optimal result \(\mathbf{X}_{p-1}^{*}\) is inspected after being used in updating the first-level approximation. The position \(i\) where \(\mathbf{\alpha}_{p-1,i}^{*}=0\) is labeled like fluorescent labeling. If some positions \(\left\{i_{1},i_{2},\cdots,i_{q}\right\}\) are labeled for \(k\) consecutive iterations, the bits of the topology variable \(\left\{\mathbf{a}_{p-1,i_{1}}^{*},\mathbf{a}_{p-1,i_{2}}^{*},\cdots,\mathbf{a}_{p-1,i_{q}}^ {*}\right\}\) are cut off from individual \(\alpha_{p-1}^{*}\) of the GA. Then, this clipped individual is used to generate the initial population for the GA process in the \(p\)-th iteration. Thus, the topology variable's dimension involved in GA optimization is reduced empirically in this way, and the search domain is shrunk at an exponential speed \(\Omega^{p}=2^{n-q}\). This will decrease the distance between approximate points and maintain a sufficiently good approximation accuracy, which offers an appropriate search direction in a large search domain. As shown schematically in Figure 6, the purple and yellow areas have the same meaning as Figure 5. The domain bounded by a red circular curve is \(\Omega^{p}\), which indicates the search space that has shrunk in each iteration. As the optimization iterates, the total search domain decreases gradually so that the bad effect on the accuracy of BMA will be weakened. With an accurate result of BMA for GA, the optimization can go along the right way and would eventually catch \(x^{*}\) in \(\Omega_{i}\) afterward.  
Figure 5: Schematic domain with iterations.  
Figure 6: Schematic domain with iterations by the LCS strategy.