## 4 Design of the POM-PSASIDEA Algorithm
### Population Size Adaptive Strategy for the Improved Differential Evolution Algorithm  
Differential evolution (DE) is an efficient global search algorithm known for its high search efficiency, strong robustness, and fast convergence speed. It is widely used in solving multi-satellite imaging task planning and multi-agent task planning problems [42; 43]. When dealing with large-scale target task scenarios, the performance of the algorithm largely depends on the population size. Too large a population size may lead to excessive consumption of computing resources, while too small a population size may lead to premature convergence of the algorithm to the local optimal solution. However, the existing DE algorithm research mainly adopts the alternate use of multiple variation strategies in the optimization process and parameter control to solve problems in the real environment, which is difficult to adapt to the solving requirements of a model at different levels, and there is a risk that feasible solutions cannot be found within the specified time limit. Therefore, this work proposes a population size adaptive strategy for the improved differential evolution algorithm (PSAIDEA) to ensure that the algorithm can solve high-quality task planning schemes while effectively improving the solution efficiency. Firstly, a population evolution state monitoring mechanism is designed to determine the adjustment of the population size to achieve balanced exploration and utilization of the DE algorithm. In the early stage of algorithm evolution, the population evolution state monitoring mechanism calculates the similarity of individuals within a population and the diversity of individuals between populations to determine whether the algorithm is facing the problem of falling into local optimality and diversity reduction to achieve a reasonable and accurate assessment of the evolution of the population. Based on the distance measure between individuals in the process of population evolution, the calculation method of population individual similarity is derived as shown in Equation (15):  
\[\gamma(x_{i},x_{j})=\frac{1}{\mu+\sqrt{\frac{\Omega}{\Omega=1}\left(x_{i},x_{ j}\right)^{2}}},x_{i}\
eq x_{j}, \tag{15}\]  
where \(x_{i}\) and \(x_{j}\) denote any two different individuals in the contemporary population, and \(\Omega\) denotes the parameter dimensions; to avoid the instability of similarity \(\gamma\), a small positive number \(\mu\) is introduced to smooth the values. Afterwards, we constructed a method for calculating individual diversity among populations, with the mathematical expression shown in Equation (16):  
\[\xi(p^{S}_{history},P_{current})=\frac{\sum\limits_{i=1}^{N^{S}_{ history}}N_{current}}{N^{S}_{history}\cdot N_{current}},\forall x_{i}\in p^{S}_{ history},x_{j}\in P_{current}, \tag{16}\]  
where \(P^{S}_{history}\) denotes the \(g\)-th iteration population in the historical environment, \(P_{current}\) denotes the current population, \(N^{S}_{history}\) denotes the number of individuals in population \(P_{history}\)\(N_{current}\) denotes the number of individuals in population \(P_{current}\), and \(x_{i}\) and \(x_{j}\) denote the individuals in population m and population \(n\), respectively. We construct the similarity matrix to measure the similarity degree among individuals in a population and store the population individual similarity calculated in each iteration into the similarity matrix as historical data. The similarity matrix constructed is shown in Equation (17):  
\[S= \begin{bmatrix}1&\gamma(x_{1},x_{2})&\dots&\gamma(x_{1},x_{n})\\ 1&1&\dots&\gamma(x_{2},x_{n})\\ \vdots&\vdots&\ddots&\vdots\\ 1&1&\dots&1\end{bmatrix} , \tag{17}\]  
Thereafter, after storing the historical data based on the similarity matrix, the Upper Quartile (UQ) is selected as the population individual similarity threshold \(Q_{\gamma}\), and all the values greater than or equal to the threshold are considered as having high similarity, while values less than the threshold are considered as having low similarity.