## 3 Methodology
### Off-Grid Arrays Design Based on GEWOA  
For off-grid arrays, considering the limit of the array element size, the minimum spacing \(d_{c}\) is constrained. This means that the search range is not the whole aperture \(L\). To solve this problem, the positions \(\mathbf{D}(d_{0},d_{1},\ldots,d_{N-1})\) are decomposed. Since the positions of the two array elements \(x_{0}\) and \(x_{N-1}\) are fixed, only the positions \(\mathbf{D}^{*}(d_{1},d_{2},\ldots,d_{N-2})\) of the \(N-2\) array elements need to be designed. Thus, a position decomposition method is applied to make \(\mathbf{D}^{*}\) continuous:  
\[\mathbf{D}^{*}=\widetilde{\mathbf{X}}+\left[ \begin{array}{c}d_{c}\\ 2d_{c}\\ \vdots\\ (N-2)d_{c}\end{array} \right]=\left[ \begin{array}{c}\widetilde{x}_{1}+d_{c} \\ \widetilde{x}_{2}+2d_{c}\\ \vdots\\ \widetilde{x}_{N-2}+(N-2)d_{c}\end{array} \right] \tag{18}\]  
In (19), \(\widetilde{x}_{1}\leq\widetilde{x}_{2}\leq,\ldots,\leq\widetilde{x}_{N-2}d_{ 1}>d_{c},d_{N-1}<L-d_{c}\). The solution for the array position \(\mathbf{D}^{*}\) is converted to the solution for \(\widetilde{\mathbf{X}}\) by the the position decomposition method. Here, the search range of \(N-2\) array elements becomes \(L-2d_{c}\). In addition, \((N-3)d_{c}\)  
Figure 4: The process of refraction learning. needs to be subtracted because the spacing between adjacent array elements is not less than \(d_{c}\). Therefore, the search aperture becomes \(L^{*}\):  
\[L^{*}=L-2d_{c}-(N-3)d_{c}=L-(N-1)d_{c} \tag{19}\]  
Through the above position decomposition, the discontinuous problem can be converted into a continuous problem. Then, GEVOA can be performed to achieve off-grid array design (Algorithm 1). ```
Input: Initial chaotic sequences \(\mathbf{S}\), maximum iterations \(t_{Max}\), and the minimum spacing \(d_{c}\) Output: Optimal individual whale position  Determine the new search aperture \(L^{*}\) using position decomposition;  Initialize \(\widetilde{\mathbf{X}}\) according to search aperture \(L^{*}\) using chaotic sequence \(\mathbf{S}\);  Find the individual position corresponding to the optimal fitness value \(\mathbf{X}^{*}\); while\(t<t_{Max}\) or \(\mathbf{PSLL}\) does not meet demanddo  Set the parameters \(\mathbf{a},\mathbf{r},l,p\), and calculate \(\mathbf{A},\mathbf{C}\). ;  Update \(\widetilde{\mathbf{X}}(t)\). if\(p<0.5\)then  if\(|\mathbf{A}|<1\)then  Enforcing shrinking encircling mechanism and updating position according to (4) and (5) ; else Random search using (8) and (9); else  Archimedes spiral strategy using (11);  end if  Set \(k\) and \(n\) and use refraction learning to find the inverse solution \(\widehat{\mathbf{X}}^{*}(t)\);  If \(\mathbf{PSLL}(\widetilde{\mathbf{X}})<\mathbf{PSLL}(\widehat{\mathbf{X}}^{*})\), choose \(\widehat{\mathbf{X}}\) as a new population; Otherwise, choose \(\widehat{\mathbf{X}}^{*}\) as a new population; \(t=t+1\)  end while
```  
**Algorithm 1**Off-grid array design based on GEVOA. The algorithmic details of the proposed method for off-grid array design based on GEVOA are summarized in Algorithm 2. First, position decomposition is performed to obtain the search range \(L^{*}\). Secondly, GEVOA is performed to search the optimal position. In the initial stage of GEVOA, the population initialization is performed by using chaos sequences to obtain \(\widetilde{\mathbf{X}}\), and the maximum number of iterations \(t_{max}\) is set. The optimal whale in the initialized population is selected by (2), and then the iterative optimization is performed. The parameters \(\mathbf{a},\mathbf{r},l,p\) of each whale are set, and \(\mathbf{A},\mathbf{C}\) are calculated for choosing different strategies of position updating. If \(p>0.5\), ASS for position updating is performed. Else, if \(|\mathbf{A}|<1\), a shrinking encircling mechanism is executed. Otherwise, a random search is used. After the position updating, the inverse solution \(\hat{\mathbf{X}}^{*}\) is generated by using RL. The best solution is selected from the new offspring and its inverse solution.