# Large-Scale Multi-Objective Imaging Satellite Task Planning Algorithm for Vast Area Mapping
## 3 Proposed Method  
### Efficient Competition Learning Particle Update Strategy  
#### 3.1.1 Improved Loser Update Strategy  
Although LMOCSO has a better convergence speed than CSO, it still requires a large number of evaluations for large-scale problems. For the LSMOP testing problem [44], the number of reference evaluations given by the author is 15,000 \(\times\) D, where D represents the number of decision variables. The computational cost is considerable. The calculation is complex and costly for the imaging satellite task planning problem, and there are many imaging strips for large areas. Therefore, obtaining good results by running LMOCSO directly to solve the imaging satellite task planning problem is difficult. To better solve the imaging satellite task planning problem, this part further improves the particle update strategy in LMOCSO.  
The core idea of the improved particle update strategy is to further accelerate the loser individual updates based on LMOCSO. The improved particle update strategy enables particles to move directly from position \(x_{l}(t)\) to position \(x_{l}(t+2)\) through only one evolution, as shown in Figure 2.  
To enable the loser to evolve from position \(x_{l}(t)\) to position \(x_{l}(t+2)\) directly, we derive the updated equation for the loser as follows:  
\[\begin{split}& V(1)=v(1)+v(2)=v(1)+r_{0}v(1)+r_{1}\Delta^{\prime}(1) \\ &=(1+r_{0})v(1)+r_{1}(X_{w}-X^{\prime}_{l}(1))\\ &=(1+r_{0})v(1)+r_{1}(X_{w}-(X^{\prime}_{l}(0)+(1+r_{0})v(1)-r_{0 }v(0)))\\ &=(1+r_{0}-r_{1}-r_{0}r_{1})v(1)+r_{1}\Delta^{\prime}(0)+r_{0}r_{ 1}v(0)\\ &=(1+r_{0}-r_{1}-r_{0}r_{1})(r_{0}v(0)+r_{1}\Delta^{\prime}(0))+r_ {1}\Delta^{\prime}(0)+r_{0}r_{1}v(0)\\ &=(r_{0}+r_{0}^{2}-r_{0}^{2}r_{1})v(0)+(2r_{1}+r_{0}r_{1}-{r_{1} }^{2}-r_{0}{r_{1}}^{2})\Delta^{\prime}(0)\\ &=R_{1}v(0)+R_{2}\Delta^{\prime}(0)\\ &\mathcal{X}^{\prime}_{l}(1)=\mathcal{x}^{\prime}_{l}(2)= \mathcal{x}^{\prime}_{l}(1)+(1+r_{0})v(2)-r_{0}v(1)\\ &=\mathcal{x}^{\prime}_{l}(0)+(1+r_{0})v(1)-r_{0}v(0)+(1+r_{0})v (2)-r_{0}v(1)\\ &=\mathcal{x}^{\prime}_{l}(0)+(1+r_{0})(v(1)+v(2))-r_{0}v(1)-r_{ 0}v(0)\\ &=\mathcal{x}^{\prime}_{l}(0)+(1+r_{0})(v(1)+v(2))-r_{0}(r_{0}v( 0)+r_{1}\Delta^{\prime}(0))-r_{0}v(0)\\ &=\mathcal{x}^{\prime}_{l}(0)+(1+r_{0})(v(1)+v(2))-(r_{0}^{2}+r_{ 0})v(0)-r_{0}r_{1}\Delta^{\prime}(0)\\ &=\mathcal{x}^{\prime}_{l}(0)+(1+r_{0})(v(1)+v(2))-R_{3}v(0)-R_{4 }\Delta^{\prime}(0)\\ \end{split} \tag{13}\]  
Figure 2: Improved particle update strategy.  
where \(V(1)\) and \(X^{\prime}_{\ l}(1)\) are the velocity vector and position vector of the updated loser, respectively. The meanings of the other symbols are the same as the above equations.  
By introducing (9) into (12) and (13) and replacing \(x^{\prime}_{\ l}(t)\) with \(x_{l}(t)\), we obtain the improved loser update strategy in ECLUS as follows:  
\[v(t+1)==R_{1}v(t)+R_{2}(x_{w}(t)-x_{l}(t)) \tag{14}\]  
\[x_{l}(t+1)=x_{l}(t)+(1+r_{0})v(t+1)-R_{3}v(t)-R_{4}(x_{w}(t)-x_{l}(t)) \tag{15}\]  
where \(r_{0}\) and \(r_{1}\) are random numbers in the range of 0 to 1, and  
\[\left\{\begin{array}{l}R_{1}=r_{0}+{r_{0}}^{2}-{r_{0}}^{2}{r_{1}}\\ R_{2}=2r_{1}+r_{0}r_{1}-{r_{1}}^{2}-{r_{0}}{r_{1}}^{2}\\ R_{3}=r_{0}^{2}+r_{0}\\ R_{4}=r_{0}r_{1}\end{array}\right. \tag{16}\]  
Therefore, the range of values for \(R_{1}\) and \(R_{3}\) is \([0,2]\), and the range of values for \(R_{2}\) and \(R_{4}\) is \([0,1]\).