# A Hybrid Optimization Algorithm for the Synthesis of Sparse Array Pattern Diagrams
## 2 Quantum Hybrid Particle Swarm Optimization Algorithm Combined with Quantum Behavior
### Quantum-Behaved Particle Swarm Optimization Algorithm
The triggering condition is that the fitness value of the particle is greater than the fitness value of the entire population, or the uniform random number is less than the basic variable. The \(P_{mutation}\) is defined as:  
\[P_{mutation}=0.2\cdot\left(1-\tanh(4\cdot\left(t/T\right))\right) \tag{15}\]  
For the selected particles to be mutated, Gaussian perturbation is performed to prevent high-fitness particles from falling into local optima, and the latest particle position is obtained as follows:  
\[ \begin{array}{l}X_{t}^{new}=X_{t-1}^{oid}+s\times\left(d_{\max}-d_{\min} \right)\times rand\\ X_{t}^{final}=\min(\max(X_{t}^{new},d_{\min}),d_{\max})\end{array}  \tag{16}\]  
The position of the particle is subjected to boundary constraints to ensure that it is within the constrained range. The boundary range is also redefined as:  
\[X_{t}^{\prime}=\left\{ \begin{array}{l l}2d_{max}-X_{t}&&X_{t}>d_{max}\\ 2d_{min}-X_{t}&&X_{t}\less d_{min}\\ X_{t}&&else\end{array} \right. \tag{17}\]  
The particle position is truncated to \(X_{t}^{\prime\prime}=\min(\max(X_{t}^{\prime}),d_{\max})\) to ensure that the search process always focuses on the effective solution space and avoids interference from out-of-bound solutions in the optimization process. The pseudo-code of the bat algorithm is shown in Algorithm 1:```
Input: Population size M, maximum generations \(T\), search space constraints: \(d_{\max}\), \(d_{\min}\), Initial parameters: \(W_{\max}\), \(W_{\min}\), \(\alpha_{0}\), \(\beta_{0}\), \(p_{m0}\) Output: the optimal result Initialization Initialize population: \(P=\{d_{i}\}_{1}^{M}\), \(d_{i}\sim U(d_{\min},d_{\max})^{N-1}\) Initialize personal bests \(\mathit{pbest}\gets p\), for\(t=1\)to\(T\)doand\(\mathit{rand}\in(0,1)\), \(\mathit{gbest}\gets\mathit{PSLL}_{\min}\) If (\(\mathit{rand}\) satisfies the first item in Equation (9)) then  The quantum gravitational center is calculated by Equation (8), and then by Equations (6), (7), (11) to (13)  The update of its particle positions mainly relies on the first term of Equation (9) Else  Update the inertia weight and learning factor through (3) to (5)  The update of the positions of its particle swarm relies on the second term in Equation (9) If (\(\mathit{rand}\) satisfies Equation (14)) then  The stop particle triggers the mutation operation through Equation (14)  The updated particle positions and boundaries are shown in Equations (15) and (16) else  Retain the population positions endif  Select the best individuals in the population and retain them for the next generation endfor  return Outputs
```

**Algorithm 1** Quantum-behaved Particle Swarm Optimization Algorithm

## 3 Experiments and Analyses

To verify the optimization performance of the proposed algorithm in sparse antenna arrays, this study conducted numerical simulation experiments on three typical array structures, namely, sparse linear arrays, planar arrays, and concentric circular arrays. In the comparative experiments, the classical particle swarm optimization (PSO), grey wolf optimization (GWO), and whale optimization algorithm (WOA) were selected as benchmark algorithms. The parameters of the comparative algorithms were all set according to authoritative references: the parameters of the PSO algorithm were set according to Reference [25], the modulation parameters of the WOA algorithm were determined according to Reference [26], and the parameters of the GWO algorithm were adopted from the recommended values in Reference [27]. To ensure the reliability of the experimental results and to fully demonstrate the superior performance of the proposed algorithm in terms of sidelobe level suppression, unified test conditions were used in the experiments: the population size was set to 50 individuals, and the maximum number of iterations was limited to 300 generations. Considering the stochastic nature of optimization algorithms, each algorithm was independently run five times, and the optimal results were selected for comparative analysis.