# Optimization Design for Sparse Planar Array in Satellite Communications
## 4 Sparse Planar Array Optimization Design Method
### Intelligent Optimization Methods
#### 4.2.3 Local Optimization Methods  
The essence of the local optimization method is based on convex optimization (CO) [65]. In 1997, [PERSON] and [PERSON] first used CO technology to solve the problem of minimizing peak sidelobe levels [66]. In array synthesis, the optimization problem can be described as a CO problem; then, the problem is essentially solved. Before introducing CO technology, convex sets and convex functions are first introduced. A convex set can be defined as a convex set for any \(x_{1},x_{2}\in C\) and \(0\leq\theta\leq 1\), if it satisfies  
\[\theta x_{1}+(1-\theta)x_{2}\in C. \tag{13}\]  
If the function \(f\) is a convex set in the domain, using \(domf\) to describe the domain, and for any \(x_{1},x_{2}\in domf\) and any \(\theta\in R\), \(0\leq\theta\leq 1\), there is  
\[f(\theta x_{1}+(1-\theta)x_{2})\leq\theta f(x_{1})+(1-\theta)f(x_{2}). \tag{14}\]  
When the inequality in the above equation holds strictly at \(x_{1}\
eq x_{2}\) and \(0\leq\theta\leq 1\), then the function \(f\) can be called strictly convex. According to the above basic concepts, we can describe the convex optimization problem as  
\[ \begin{array}{c}\min f(x)\\ s.t.\ \ g_{i}(x)\leq 0,i=1,2,\ldots,m\\ h_{i}(x)=0,i=1,2,\ldots,p,\end{array}  \tag{15}\]  
where \(f\) and inequality \(g_{i}\) are convex functions, and \(h_{i}\) is a radiative function. There are three main standard forms of convex optimization, as shown in Figure 14. Sparse area array synthesis problems can be converted into these standard forms and then solved using toolkits such as CVX [67], and SeDuMi [68]. At present, the second-order cone program (SOCP) is the most widely used in sparse area matrix synthesis [69]. For example, the authors of [70] proposed to transform the array pattern synthesis problem into an SOCP problem, and then synthesize the pattern with very low peak sidelobe level using the cyclic iteration method under the condition of satisfying the expected main lobe, which is suitable for any array. At present, local optimization methods have been widely used in sparse array optimization design. Aiming at the beamforming problem, an iterative beam synthesis method based on sequence convex optimization was proposed in [71], which realizes the synthesis of arbitrary-shaped beams and the minimization of peak sidelobe levels. Aiming at the array synthesis problem, the authors of [72] proposed an array synthesis method based on convex optimization, which uses conjugate symmetric beamforming technology to form weights, so that the upper and lower bounds of the antenna pattern are constrained as convex. The problem is transformed into a second-order cone planning problem. A coupling matrix is introduced to reduce the coupling effect between the elements, the main lobe of any beam width and response ripple can be obtained by using a smaller number of elements, and the designed element spacing can be greater than half wavelength, reducing the coupling effect without the presence of the grating lobes. This method has good robustness. Aiming at the synthesis of ultrasparse array patterns in large aperture scenes, a fast algorithm using the alternating direction multiplier method was proposed to optimize the position and weight of the array element and effectively reduce the peak sidelobe level [73]. An iterative  
Figure 14: Common standard form of convex optimization. scheme was used to propose a sparse area array synthesis method based on sequence convex optimization, reducing CPU time consumption while reducing the number of elements and peak sidelobe level [74]. In a follow-up study, the author proposed a reconfigurable sparse array synthesis method based on sequence convex optimization [75], which simultaneously optimizes the number of elements and excitation weights and can achieve an arbitrary beam width and secondary lobe level, with good robustness. The authors of [76] proposed a joint (JCO) integrated approach.