## 4 Fibonacci Tree Optimization Algorithm  
Recall that one-dimensional Fibonacci (golden ratio) search cannot proficiently take care of multi-variate issues, giving rise to Fibonacci Tree Optimization--an improved multi-dimensional search algorithm [44]. Let \(\mathbf{X_{A}}\), \(\mathbf{X_{B}}\) and \(\mathbf{X_{C}}\) be the vectors in D-dimensional Euclidean space. In particular, \(D=2\). \(\mathbf{X_{A}}\) and \(\mathbf{X_{B}}\) address the endpoints of the search component fulfilling the optimization rule, and \(\mathbf{X_{C}}\) signifies the split points that can be resolved from searching rules. A proportion of the vectors can be determined as follows:  
\[\frac{\left\|\mathbf{X_{C}}-\mathbf{X_{A}}\right\|}{\left\|\mathbf{X_{B}}- \mathbf{X_{A}}\right\|}=\frac{\left\|\mathbf{X_{B}}-\mathbf{X_{C}}\right\|}{ \left\|\mathbf{X_{C}}-\mathbf{X_{A}}\right\|}=\frac{F_{p}}{F_{p+1}}, \tag{28}\]  
where \(F_{p}\) denotes the \(p\)th Fibonacci number, and \(\frac{F_{p}}{F_{p+1}}\) is equal to what is called the golden ratio. The fitness function determined by the endpoints in the construction ought to be assessed as:  
\[J(\mathbf{X_{A}})<J(\mathbf{X_{B}}). \tag{29}\]\(J(\cdot)\) denotes the objective function. Then, the coordinate for split point \(\mathbf{X_{C}}\) can be calculated as:  
\[\mathbf{X_{C}}=\mathbf{X_{A}}+\frac{F_{p}}{F_{p+1}}(\mathbf{X_{B}}-\mathbf{X_{A} }). \tag{30}\]  
Finding the optimal value can likewise be viewed as setting up a search component in FTO, and can be partitioned into two phases: the local optimization phase and the global optimization phase. Allow \(\Theta\) to signify the points set of the objective function and set \(\left|\Theta\right|_{num}=F_{p},i=1,2,\cdots,n\), where \(\left|\cdot\right|_{num}\) denotes the number of points in the set, and \(d\) represents the depth of the tree. We select the point with the best fitness value. Then, in the following optimization stage, the points are rearranged by their fitness values from best to worst. There are two search rules in FTO, which can be summed up:  
**Rule One**: Consider the endpoints \(\mathbf{X_{A}}\) and \(\mathbf{X_{B}}\), which are given by  
\[\{\mathbf{X_{A}}\}=\Theta=\left\{\mathbf{X_{k}}|k\in[1,F_{p}]\right\} \tag{31}\]  
\[\{\mathbf{X_{B}}\}=\Theta=\left\{\mathbf{X}|\mathbf{X}\in\prod_{j=1}^{D}\left[ X_{lb}^{I},X_{ub}^{I}\right]\right\}, \tag{32}\]  
where \(\Theta\) denotes the points set, with each point represented as \(\mathbf{X_{k}}\), of the \(p\)th iteration, and \(k\) is the index. \(\mathbf{X_{A}}\) absorb every point from \(\Theta\). \(\mathbf{X_{B}}\) select \(F_{p}\) points from \(\Theta\) randomly, where \(F_{p}\) denotes the population size. \(X_{ub}^{I}\) and \(X_{lb}^{I}\) are the upper and lower limits for every point. For \(\forall\mathbf{X}\in\{\mathbf{X_{B}}\}\), \(\mathbf{X}\) satisfies a uniform distribution over the span \(\left[X_{lb}^{I},X_{ub}^{I}\right]\), viz. \[\Pr(\mathbf{X})=U(X_{lb},X_{ub})=\frac{1}{X_{ub}-X_{lb}}. \tag{33}\]  
Using the \(\mathbf{X_{A}}\) and \(\mathbf{X_{B}}\) given above, the split points \(\mathbf{X_{S1}}\) are solved by Equation (30). **Rule Two**: Assume that \(\mathbf{X_{best}}\) is the optimal point in the current iteration given by rule one, viz.