## 2 Problem Description  
Let us consider that a sparse linear array has \(N\) isotropic radiative elements, which are distributed along the \(x\)-axis at \(\mathbf{D}=[d_{0},d_{1},\ldots,d_{i},\ldots,d_{N-1}],i\in[0,N-1]\). The array aperture is \(L\). The beampattern of the array is given by:  
\[P=\sum_{i=1}^{N}w_{i}\mathrm{e}^{\frac{\mathrm{j}\pi}{\lambda}d_{i}\sin( \theta-\theta_{0})} \tag{1}\]  
where \(\mathrm{j}=\sqrt{-1},\lambda\) denotes the wavelength of the operating frequency, \(w_{i}\) denotes the complex excitation of the \(i\)th element in the array, \(\theta\) is the steering angle, and \(\theta_{0}\) representsthe desired beam direction. The schematic of the array elements distribution is shown in Figure 1. Sparse arrays can be divided into on-grid arrays and off-grid arrays according to the positions of the array elements. The grid points are set at positions that are integer multiples of half the wavelength.  
The black solid circles in Figure 1 are the actual positions of the array elements. The red hollow circles are grid points. If all array elements fall on the grid points, it is an on-grid array. Otherwise, it is an off-grid array. The PSLL of the off-grid array could be lower than that of on-grid array. In other words, an off-grid array can use fewer elements than an on-grid array to meet the same PSLL. However, the on-grid array is easier to process with less computation charge. Therefore, different design methods can be selected with a trade-off between the PSLL and computation charge.  
In this paper, we focus on the layout problem of large-scale sparse sensor arrays, and all weights are set to 1 (\(w_{n}=1\) for all elements). The beampattern of a uniform array is a Dirichlet function with a PSLL of about \(-13\) dB. However, the positions of the sparse array can be distributed nonuniformly. Assuming that the weight vector of a uniform array is \(\hat{\mathbf{w}}\), then the weight vector \(\mathbf{w}\) of the sparse array can be expressed as \(\mathbf{w}=\hat{\mathbf{w}}\odot\mathbf{t}\), where \(\mathbf{t}=[1,0,1,\ldots,0,1]\) can be considered a tapering vector, which consists of 1 and 0, with 1 for element and 0 for no element, and \(\odot\) denotes the Hadamard product.  
The tapering operation enables the side lobes to be lower than those of the uniform array. The optimal positions by minimizing PSLL are filled with 1 in vector \(\mathbf{t}\). The beampattern of a sparse array obtained through Equation (1) is the convolution of the Dirichlet function with the spatial spectrum of weight vector \(\mathbf{w}\), which is based on the array positions. Hence, a sparse sensor array with a low PSLL can be obtained by optimizing the array element positions. Considering the physical limit of the sensors' size, the minimum spacing of the elements must be restricted. Therefore, the array design problem can be formulated as follows:  
\[\min_{\mathbf{X}}\mathbf{PSLL}(P)\quad\text{ s.t. }\left\{\begin{array}{c}x_{i}-x_{j} \geq d_{c}>0\\ i,j\in\mathbf{Z},0\leq j\leq i\leq N-1\\ x_{0}=0,x_{N-1}=L\end{array}\right. \tag{2}\]  
where \(x_{i}\) and \(x_{j}\) denote the position of the \(i\)th and \(j\)th element, respectively. \(d_{c}\) is the minimum distance between the adjacent elements. \(\mathbf{Z}\) denotes the set of positive integers.