## 2 Formulation and Algorithm  
In this section, the design is discussed first. Then, mathematical modeling is carried out to represent the object. Finally, the branch and bound technique with convex optimization used to solve the problem is outlined. 1. Problem Statement  
The full array has 64 \(\times\) 64 elements and is shown in Figure 1. The element spacings between each element is a half-wavelength at the center frequency. For the thinned array, only 2400 elements of the full array can be selected. Also, as shown in Figure 1, every 2 \(\times\) 2 element as a subarray is deployed for a single transceiver. The thinned array selection is based on each 2 \(\times\) 2 element. For the array radiation pattern, in the \(\varphi\) = 0\({}^{\circ}\), \(\varphi\) = 45\({}^{\circ}\), \(\varphi\) = 90\({}^{\circ}\), and \(\varphi\) = 135\({}^{\circ}\) cut planes, the array pattern in the sidelobe region (\(h(\theta,\varphi)\)) should satisfy the following requirements:  
\[h(\theta,\varphi)\leq \begin{cases}-9-25\times\log_{10}(\theta)\ dBi,&2^{ \circ}\leq\theta\leq\gamma^{\circ}\\ -30\ dBi,&7^{\circ}\leq\theta\leq 9.2^{\circ}\\ -6-25\times\log_{10}(\theta)\ dBi,&9.2^{\circ}\leq\theta\leq 48\\ -36\ dBi,&48^{\circ}\leq\theta\leq 90^{\circ}\end{cases} . \tag{1}\]  
Also, for the symmetric of the array radiation pattern, most of the subarrays should line in the circle. 1. Mathematical Modeling  
A 2 \(\times\) 2 subarray was designed for the full 64 \(\times\) 64 array, where the sidelobe constraints are \(h(\theta,\varphi)\). The binary integer matrix \(\mathbf{B}\in\mathbb{R}^{32\times 32}\) was defined as the optimization variable. Here \(B_{ij}=1\) means that 2 \(\times\) 2 subarray was placed and \(B_{ij}=0\) means that 2 \(\times\) 2 subarray was not placed at one of the four positions \((2i-1,2j-1)\), \((2i-1,2j)\), or \((2i,2j-1)\). Therefore, the selection matrix \(B\bigotimes l_{2\times 2}\) was achieved for the full array, where \(\bigotimes\) denotes the Kronecker product and \(l_{2\times 2}\) denotes the \(2\times 2\) matrices. Based on the construction, the constraints were transformed into a mathematical problem. Meanwhile, to ensure that most subarrays lie within the circle, the subarrays outside the circle were inactive, i.e., \(B_{ij}=0\), \((i,j)\in\Phi\), where \(\Phi\) denotes the set of indices for the subarrays outside the circle. For the full 64 \(\times\) 64 array, \(\theta\) and \(\varphi\) denote the azimuth and elevation angle, and the response of the (\(m\),\(n\))\(th\) array element is \(a_{m,n}(\theta,\varphi)=e^{j((m-1)\tau cos(\varphi)sin(\theta)+(n-1)\tau sin( \varphi)sin(\theta))}\). Therefore, given the distribution of the array, the array radiation pattern \(f(\theta,\varphi)\) can be obtained as  
\[ \begin{array}{l}f(\theta,\varphi)=\sum\limits_{m=1}^{64}\sum\limits_{n=1}^{ 64}(a_{m,n}(\theta,\varphi)\times 0.5(1+\cos(\theta)))[\textbf{B}\otimes 1_{2\times 2 }]_{m,n}\\ a_{m,n}(\theta,\varphi)=e^{j((m-1)\tau cos(\varphi)sin(\theta)+(n-1)\tau sin( \varphi)sin(\theta))}\end{array}  \tag{2}\]  
Consequently, finding the selection matrix **B** satisfying the sidelobe constraint can be formulated as the following optimization problem  
\[ \begin{array}{l}find\ \textbf{B}\\ \text{s.t.}\qquad f(\theta,\varphi)=\left|\sum\limits_{m=1}^{64}\sum\limits_{n= 1}^{64}(a_{m,n}(\theta,\varphi)\times 0.5(1+\cos(\theta)))[\textbf{B}\otimes 1_{2 \times 2}]_{m,n}\right|\leq h(\theta,\varphi)\\ \qquad\qquad B_{i,j}=0,\ (i,j)\in\Phi\\ \qquad\qquad\sum\limits_{i=1}^{32}\sum\limits_{j=1}^{32}B_{i,j}=Q\\ \qquad\qquad\textbf{B}\in\{0,1\}^{32\times 32}\end{array} . \tag{3}\]  
where \(h(\theta,\varphi)\) is the requirements of sidelobe level in Equation (1), and \(Q\) is the number of the thinned array elements which should be active. Problem Solving  
It should be noted that the number of binary variables is 32 \(\times\) 32 = 1024, where a binary variable of 1 indicates the deployment of the \(2\times 2\) subarray.