## Embedded Ordinal Optimization into Tree-Seed Algorithm
### Improved Tree-Seed Algorithm  
Next, we can efficiently select \(N\) outstanding solutions from search space using optimization approaches assisted by the RMTS surrogate model. Because the TSA iteratively aims at improving the candidate solutions, it would be best suited for our needs. The advantages of TSA include ease of implementation, only a few parameters need to be adjusted, tactical interplay between searching diversification and intensification, quick convergence and high efficiency [14]. TSA is inspired from the relation between trees and seeds, which simulates the growth cycles of trees and seeds on a land [16]. The trees and their seeds correspond to the possible solution while the land indicates the search space of the optimization problems. At the beginning of the iteration, the trees are sowed to the land. Then, a pre-defined number of seeds for each tree are generated during the search process. If the performance of the best seed is better than its tree, the best seed is added to the stand and its tree is removed from the stand. This search process is performed until a required number of iteration is met. The following is a description of the developed ITSA. Firstly, an initial tree population is generated. Each tree is randomly assigned an initial position. Secondly, a new seed is spread from a tree, either the best tree location of another tree location is considered with its own location. The first scheme uses the best tree to provide a speed convergence, and the second one uses the randomly selected tree to explore spacious regions on the search space. The key point is which tree location will be chosen to generate a new seed location. This selection depend on the search tendency (\(ST\)), which varies within a range of 0-1. The number of seeds generated by a tree can be more than one, which depends on the seed produced rate (\(SPR\)). More seeds improve the global search capability but require more computing time. This iterative process is performed until the pre-defined number of iterations has been reached. The ITSA has two control factors: \(ST\) and \(SPR\). The intensification and diversification of the ITSA are mainly controlled by the \(ST\). A large \(ST\) causes a speed convergence and powerful local search, while a small \(ST\) provides a powerful global search but slow convergence. When the \(ST\) is decreased, the ability of local search decreases and the ITSA trends to perform the global search. When the \(ST\) is increased, the ITSA tends to conduct the local search around a local region. Similarly, a lower \(SPR\) tends to local search, while a higher \(SPR\) leads to global search. The algorithmic notations used in the ITSA are described as follows. \(\Psi\) denotes the size of tree population; \(k_{\max}\) denotes the pre-defined number of iterations; \(ST^{k}\in[ST_{\min},ST_{\max}]\) is the \(ST\) at iteration \(k\), where \(ST_{\max}\) and \(ST_{\min}\) represent the maximum and minimum of \(ST\), respectively;  
Figure 1: Structure of a regularized minimal-energy tensor-product splines. and \(\gamma^{k}\in[\gamma_{\min},\gamma_{\max}]\) is the \(SPR\) at iteration \(k\), where \(\gamma_{\max}\) and \(\gamma_{\min}\) denote the maximum and minimum of \(SPR\), respectively. The position of the \(i\)th tree and the \(j\)th seed at iteration \(k\) in an \(n\)-dimensional space is represented by \(\mathbf{x}_{i}^{k}=[x_{i,1}^{k},\ldots,x_{i,n}^{k}]^{\mathrm{T}}\) and \(\mathbf{s}_{j}^{k}=[s_{j,1}^{k},\ldots,s_{j,n}^{k}]^{\mathrm{T}}\), respectively. According to the analysis mentioned above, the ITSA can be summarized as follows. ```
Step 1: Define parameters Define the values of \(\mathbf{v}\), \(k_{\max}\), \(ST_{\min}\), \(ST_{\max}\), \(\gamma_{\min}\), \(\gamma_{\max}\), and set \(ST^{0}=ST_{\min}\), \(\gamma^{0}=\gamma_{\max}\), \(k=0\), where \(k\) denotes the iteration index. Step 2: Initialization (a) Initialize the first population of \(\Psi\) trees with positions \(\mathbf{x}_{i}^{k}\) as follows.