## 2 Materials and Methods
### Adaptive DBO for DPE
#### 2.3.2 Adaptive DBO Algorithm
Thus, it is advantageous if the mixed strategy coefficients related to the given local fitness landscape can be self-determined. Fitness landscapes bridge algorithms and their optimization problems, with ruggedness describing the complexity of these landscapes. To calculate the information entropy ruggedness, start with a population \(P\) consisting of \(L\) individuals, corresponding to a fitness value vector \(\mathbf{f}\). Then, derive the differential sequence \(\Delta F=[\Delta f_{1},\cdots,\Delta f_{L-1}]\) by  
\[\Delta f_{l}=f_{l+1}-f_{l},l=1,\cdots,L-1. \tag{25}\]  
Define a symbolic function with respect to \(\epsilon\) as  
\[s(x)=\left\{ \begin{array}{ll}\mathbb{I},&x<-\epsilon\\ 0,&-\epsilon\leq x\leq\epsilon\\ 1,&x>\epsilon.\end{array} \right. \tag{26}\]  
We obtain the sequence \(S(\epsilon)=[s(\Delta f_{1}),\cdots,s(\Delta f_{L-1})]\). The parameter \(\epsilon\) is a constant, where a smaller \(\epsilon\) increases sensitivity to the differences between neighboring fitness values. Figure 3: Comparison of improved and original convergence factors \(R\) under different optimization scenarios. The meanings of consecutively occurring symbols in \(S(\epsilon)\) are listed in Table 1. The entropic measure, based on the frequency of different symbol combinations in \(S(\epsilon)\), is defined as  
\[H(\epsilon)=-\sum_{p\
eq q}P_{[pq]}\log_{6}P_{[pq]}, \tag{27}\]  
where \(P_{[pq]}=n_{[pq]}/(L-1)\) is the frequency of two different sign values \(p,q\in\{\bar{1},0,1\}\) occurring consecutively in \(S(\epsilon)\), and \(n_{[pq]}\) is the count of sub-blocks \(pq\) in \(S(\epsilon)\). The logarithm base is six due to the six possible rugged shapes in Table 1, making \(H(\epsilon)\) a _b-ary_ entropy measure ranging from 0 to 1. This measure indicates landscape complexity by quantifying the frequency of consecutive combinations of different symbol values in \(S(\epsilon)\). The entropy is a measure of the amount of information, so the larger \(H(\epsilon)\) is, the greater the variety of combinations of consecutively different sign values in the sequence \(S(\epsilon)\), and the more structure and ruggedness that the landscape contains. The smaller \(H(\epsilon)\) indicates that the sequence \(S(\epsilon)\) has a more homogeneous variety of combinations of consecutively different sign values, and the landscape contains less and smoother structure. For the same difference sequence \(\Delta F\), the measurement of its ruggedness is influenced by the size of \(\epsilon\), which determines the interval of the neutral region. The larger \(\epsilon\) is, the larger the interval of the neutral region, equivalent to an observer viewing the local landscape from a distance, where many tiny undulations cannot be observed. In this case, \(H(\epsilon)\) provides a coarse-grained observation. Conversely, the smaller \(\epsilon\) is, the closer the observer is to the local landscape, allowing for the observation of any small changes in the fitness values. Here, \(H(\epsilon)\) provides a fine-grained observation. The smallest value of \(\epsilon\) for which the landscape becomes flat, i.e., for which \(S(\epsilon)\) is a string of zeros, is called the information stability measure and is denoted by \(\epsilon^{*}\)[35]. The value of \(\epsilon^{*}\) would therefore equate to the largest difference in fitness values between any two successive points on the path. Let \(\epsilon\) be a set of values that gradually decrease, approaching zero, referred to as the information entropy ruggedness vector \(\mathbf{H}(\epsilon)\).