## 4 Fibonacci Tree Optimization Algorithm
\[\mathbf{X_{best}}=BEST(\Theta). \tag{34}\]  
Then, let \(\mathbf{X_{A}}=\mathbf{X_{best}}\); we have  
\[J\{\mathbf{X_{A}}\}=\min\{J\{\mathbf{X_{k}}\}|k\in[1,F_{p}]\} \tag{35}\]  
\[\mathbf{X_{B}}=\{\mathbf{X_{k}}|\mathbf{X_{k}}\in\Theta\wedge\mathbf{X_{k}} \
eq\mathbf{X_{A}}\}. \tag{36}\]  
Hence, the split points \(\mathbf{X_{S2}}\) in the local optimization stage can be determined by Equations (35```
Input: Branch depth: \(d\); Max iterations: \(I_{max}\) Output: Optimal point: \(\mathbf{X_{best}}\)
1 Initialize set \(\Theta\) with \(F_{1}\) global random points;
2 for\(p=1\to I_{max}\)do
3 Rule 1: \(\{\mathbf{X_{A}}\}\leftarrow\Theta\);
4 Generates \(\{\mathbf{X_{B}}\}\), \(|\{\mathbf{X_{B}}\}|_{num}=F_{p}\);
5 Compute the split points set \(\{\mathbf{X_{S1}}\}\) and merge them into the set \(\Theta\);
6 Sort and preserve top best \(F_{p}\) points of the set \(\Theta\);
7 Rule 2: \(\mathbf{X_{A}}\leftarrow\mathbf{X_{best}}\);
8\(\{\mathbf{X_{B}}\}=\{\mathbf{X_{k}}|\forall\mathbf{X_{k}}\in\Theta\wedge \mathbf{X_{k}}\
eq\mathbf{X_{A}}\}\);
9 Compute the split points set \(\{\mathbf{X_{S2}}\}\) and merge them into the set \(\Theta\);
10 Sort and preserve top best \(F_{p}\) points of the set \(\Theta\);
11\(p\gets p+1\);
12 if\(|\Theta|_{num}=F_{d}\)then
13 break;
14 end if
15
16 end for return \(\mathbf{X_{best}}\);
```

**Algorithm 1**Pseudo-code of Fibonacci branch generation process

## 5 Analysis

### Why FTO Algorithm Is Chosen for Parameter Optimization? 1. In the FTO algorithm, the growth of a Fibonacci tree is based on the optimal points of the latest generation, which is a process of competitive elimination. If there are peaks with different heights, the smaller peaks will be removed in the optimization process. Hence, the global optima can be achieved. 2. The inner structure of the FTO algorithm meets the golden ratio all over. With each iteration, the separation of the golden ratio rapidly compresses the search space,

Figure 5: The diagram of the branch generation process in the Fibonacci tree optimization algorithm. providing the local optimal solution. It is adapted to the optimization of multi-modal functions. Conversely, most heuristic algorithms are based on the probabilistic search; they are essentially trial and error methods. It may take thousands or even millions of iterations, which is uneconomical and slow. * The FTO setup only concerns the initial Fibonacci number, which is exquisite and simple. The typical way to obtain multiple local extremums of the multi-modal function using heuristic algorithms is the niche technology. Setting population parameters has a significant impact on the optimistic effects. * Just like most heuristic algorithms, FTO is also inspired by natural phenomena--plant photoptopism. Plants flourish more in sunny areas. To solve the optimization of the multi-modal function, FTO considers the global search space as a sunlit area. The hit probability is sustained and stable. The lush areas refer to the local search process, reflecting the fact that FTO can alternate between global and local searches. * It fully utilizes the computer memory to save the optimization process and therefore is traceable. The comparison between FTO and other optimization algorithms is listed in \[J(\mathbf{X_{A}})<J(\mathbf{X_{B}}). \tag{29}\]0 Figure 5(a) illustrates the idea of golden separation in the FTO algorithm's structure. Figure 5(b) reflects the global/local alternate search process inspired by phototropism of plants. \begin{table}
\begin{tabular}{l l l} \hline \hline  & **FTO** & **Heuristic Algorithms** \\ \hline Parameters & simple & many, hard to determine \\ Imitation of nature & phototropism & swarm intelligence \\ Characteristic & competitive elimination alternate & random search based on \\  & search golden ratio search & probability \\ Global search capability & strong & prone to be trapped in local \\ Convergence speed & fast & slow \\ Resources footprint & low & high \\ \hline \hline \end{tabular}
\end{table}