Differential evolutionary algorithm based on principal component analysis for the satellite bandwidth resource scheduling problem

This paper focuses on the research of the communication satellite bandwidth resource scheduling problem. Satellite resource scheduling refers to adjusting the order of task execution within a certain scheduling time to complete as many satellite tasks as possible while satisfying bandwidth and time constraints. Traditional algorithms cannot meet the time requirements in the process of satellite communication resource scheduling. Therefore, this paper proposes a differential evolution algorithm based on principal component analysis (PCA) and combines it with the Lowest Horizontal Line Algorithm (LHLA) to solve this problem. The method was validated on two datasets at 20 dimensions and 40 dimensions, and compared with other traditional DE algorithms. The experimental results showed that the method obtained better solutions.


Introduction
The Rectangle Packing problem is an NP-hard problem that requires the use of heuristic algorithm optimization algorithms to solve.The goal of this problem is to place a series of rectangles with different sizes in a rectangle frame so as to maximize the utilization of the frame area and ensure that the rectangles do not overlap with each other.Satellite communication problems [1] can be modeled using rectangle packing problems.Each task is abstracted as a rectangle, and the length and width of the rectangle, respectively, represent the required time and bandwidth of the task.
In recent years, many researchers have addressed the Rectangle Packing problem by using various optimization algorithms [2].Due to the complexity of the Rectangle Packing problem, these algorithms cannot guarantee to obtain the optimal global solution.They can only obtain local optimal or suboptimal solutions, which has resulted in the inability to meet the rapid calculation and performance requirements in satellite communication application scenarios.Thus, this paper proposes a Main Component Differential Evolution (MCDE) algorithm to solve the Rectangle Packing Problem (RPP) [3], which is an optimization algorithm that reduces high-dimensional vectors to low-dimensional through Principal Component Analysis (PCA) to compress search space with minimal information loss.The MCDE algorithm analyzes the main direction of the current population.It determines effective search directions and speeds up the search process.During this process, the MCDE algorithm also considers the contribution rate of the variance of the principal component to ensure the performance of the search direction.Compared with the DE (Differential Evolution) algorithm, the MCDE algorithm has higher search speed and better optimization performance and can avoid getting stuck in local optimal solutions [4].Therefore, we use the MCDE method to attempt to solve the rectangle packing problem in the condition of satellite communication resource scheduling.

Modeling of satellite communication resource scheduling
The satellite communication resource scheduling problem refers to how to allocate a certain amount of satellite bandwidth resources to multiple communication tasks within a continuous period of time.The time and frequency band of these communication tasks are non-overlapping, and the required bandwidth resources must be continuous.Due to the different duration and required bandwidth of the tasks, there may be unused bandwidth resources after the task is completed, which will lead to resource fragmentation.Therefore, a reasonable scheduling scheme is needed to minimize the fragmentation of bandwidth resources and maximize the utilization of satellite resources.As shown in Figure 1, the satellite communication resource scheduling problem can be abstracted by representing each communication task as a rectangle and arranging these rectangles in a twodimensional space [5].In this rectangle packing problem, the duration of the task corresponds to the width of the rectangle, and the required bandwidth resource corresponds to the height of the rectangle.Since the time and frequency bands of the task cannot overlap, these rectangles cannot overlap in the two-dimensional space of bandwidth and time.The height of the two-dimensional space [6] is equal to the total satellite bandwidth resources, and the width of the two-dimensional space is equal to the sum of all task durations.Therefore, this problem can be regarded as a rectangle packing problem [7].

Constraint condition
Given a set of n mutually independent Rectangles M , … , M to be packed into a two-dimensional space of width B and height H [8], the objective is to find the optimal packing sequence based on the Lowest Horizontal Line Algorithm to maximize space utilization.As shown in Figure 2.

M-packing rectangle (Communication mission);
H min -minimum bandwidth after sorting; B-the known time limit for a certain spacecraft; H-the upper limit of the known bandwidth for a certain spacecraft; H i , B i -the height and width span of rectangle Mi (Resource bandwidth and time); (x i l , y i l )-the coordinate of the bottom-left corner of the rectangle M i ; (x i r , y i r ) -the coordinate of the top-right corner of the rectangle M i ; C ij =0 or 1: When Mi is located above M j , the logical variable is set to 1; when it is located below, the variable is set to 0.D ij =0 or 1: When Mi is located to the left of M j , the logical variable is set to 1; when it is located to the right, the variable is set to 0.
The constraint formula is as follows: ,  ,  ,  0 C , D ∈ 0, 1 Formulas ( 1) and ( 2) constrain the width and height ranges of the matrices; Formulas (3) and (4) determine the geometric relationship between the lower left and upper right coordinates of the rectangle be allocated; Formula (5) represents the adjacency restriction of the width and height of the rectangle when M i and M j are horizontally adjacent; Formula (6) represents the adjacency restriction of the width and height of the rectangle when M i and M j are vertically adjacent; Formula (7) constrains the position relationship between Mi and M j to be either horizontal or vertical, ensuring that the matrices are independent and do not overlap with each other.

Objective function
The objective function is an important criterion for evaluating the adaptability of chromosomes in the population, determining whether the chromosome sequence is retained for the next round of crossover and mutation.The larger the corresponding fitness value of the population is, the more optimal the population is.Based on the rectangle dataset and constraints, the rectangle packing problem [9] can be expressed as a specific profit formula as follows (10):

PCA searching operator
The MCDE algorithm first randomly initializes a population pop and then iterates maxiter times for optimization.In each iteration, the fitness of the population is first calculated, and the population is sorted according to fitness, selecting the top k populations with the largest/smallest fitness.Then, the principal component analysis (PCA) method is used to calculate and sort the principal component vectors using the fit_transform function, retaining the top n principal component vectors and reconstructing the projection points in the original search space.Next, new individuals are generated using mutation and crossover operations.Finally, the selection operation is used to update the population, reaching the iteration limit and returning the optimal solution.The Main Direction proposed in this paper is as follows: Step 1.The raw data is preprocessed by removing the mean.The n×1 mean vector m and the covariance rectangle  are calculated as follows ( 11 where k is the dimensionality of the reduced space. Step 3. x i is projected onto the m-dimensional space, the formula as shown in (13).
(13) Step 4. The projected point x i is reconstructed in the original space, the formula as shown in ( 14). (14)

The overall flowchart of the MCDE
Firstly, the population is initialized, and the PCA search operator is used to extract the principal component direction to reconstruct the population.Then, the DE algorithm is used to optimize the sequence of rectangular packing [10], and the lowest horizontal line method is adopted to achieve the packing of the rectangular.A complete rectangular packing optimization algorithm can be constructed, as shown in Figure 3.

Experimental settings
The parameters of MCDE and DE are set as follows.The setting is chosen as the same as that used in DE for a fair comparison.Experimental environment: Windows 11 system and Pycharm software.Basic parameter settings: population N=30, packing space X-axis length B=30, Y-axis length H=100, scaling factor F=0.5, crossover probability C=0.8, and reduce dimension to component =10.

Comparison Experiment 1 Between MCDE and DE
In order to verify the performance of optimization improvement on two-dimensional rectangle packing utilization, the comparison between the MCDE algorithm and the original DE algorithm in optimizing improvement in high-dimensional and low-dimensional population dimensions is shown in Figure 4 and Figure 5.We can see that in the sets of 20 and 40 packing rectangles, the convergence speed of DE is faster than MCDE when the number of iterations is low.Still, in high dimensions of 40 packing matrices, MCDE can escape local optima and find a better global solution as shown in Table 2.

Comparison Experiment 2 between MCDE and DE
To evaluate the performance of MCDE, the three datasets of 20 and 40 dimensions were randomly generated.Two algorithms were used to independently run 20 times on each of the three datasets, and the average values were taken.As shown in Table 1, MCDE can obtain better results within a certain number of iterations.This is because the MCDE algorithm can better capture the main features in the raw data and generate better packing results.However, because the MCDE algorithm needs to update multiple components in each iteration, the computational complexity is higher, and the running time of the MCDE algorithm is longer than that of the DE algorithm.
The specific meanings of the data are the following.U min : the minimum utilization rate.U avg : the average utilization rate.U best : the best utilization rate.T avg : the average running time of the algorithm.

Conclusions
In this paper, we used MCDE to further optimize the Rectangle Packing Problem based on the Lowest Horizontal Line Algorithm.Based on the comparison experiments with DE, the conclusions of this paper can be summarized as follows.Under the same number of iterations, the DE algorithm can quickly converge to the local optimal solution.However, as the number of iterations increases, it is apparent that the MCDE algorithm finds a higher fitness value than the DE algorithm.This is because the MCDE algorithm requires updating multiple components in each iteration, leading to higher computational complexity.From the perspective of packing quality, the MCDE algorithm is superior to the DE algorithm.This is because the MCDE algorithm can better capture the main features in the original data, thereby generating better packing results.The DE algorithm has better stability compared to the MCDE algorithm.The MCDE algorithm is prone to algorithm degradation on certain datasets, while the DE algorithm is relatively stable.Therefore, when using DE and MCDE algorithms for packing optimization, suitable algorithms can be selected based on specific application scenarios and optimization goals to achieve better optimization results.

Figure 2 .
Figure 2. Coordinate schematic diagram of rectangle packing in space.

2 .
The eigen decomposition on the covariance rectangle is performed to obtain the eigenvalues and their corresponding eigenvectors   , … ,   .The feature vectors are sorted according to their corresponding eigenvalues, and the top   , … ,   feature vectors are selected as principal components,

Figure 4 .
Figure 4. Comparison of convergence curves in 20 dimension.

Figure 5 .
Figure 5.Comparison of convergence curves in 40 dimension.

Table 1 .
compare the performance of MCDE and de in different dimensions.

Table 2 .
average utilization of MCDE and de at different iterations performance.