A dynamic population multi-objective particle swarm optimization method based on crowding degree and contribution degree

Abstract

The application relates to a dynamic population multi-objective particle swarm optimization method based on crowding degree and contribution degree, and relates to the technical field of intelligent optimization algorithms, in particular to a dynamic population multi-objective particle swarm optimization method based on crowding degree and contribution degree. The method aims to solve the problem that the calculation efficiency and solution set quality of a traditional multi-objective particle swarm optimization (MOPSO) algorithm cannot be considered simultaneously when solving complex engineering problems. The method establishes grid division to realize solution space discretization, combines non-dominated sorting and Euclidean distance to calculate particle contribution degree, and defines a crowding degree index based on the total number of particles in the grid. Then, a particle deletion probability formula is constructed based on the contribution degree and the crowding degree, a dynamic deletion strategy and a quantity limitation mechanism are introduced, and low-value and high-redundancy particles in the population are adaptively removed according to the deletion probability, so that the population size is reduced and the calculation efficiency of the algorithm is improved under the premise of ensuring the diversity and convergence of the population. The method provided by the application has obvious advantages in efficiently solving multi-objective optimization problems in combination with non-dominated sorting, grid division and a dynamic deletion strategy.

Description

Technical Field

[0001] This invention belongs to the field of interdisciplinary application technology of computational mathematics and intelligent computing, and relates to intelligent optimization algorithms, specifically a dynamic population multi-objective particle swarm optimization method based on crowding and contribution. Background Technology

[0002] Multi-objective particle swarm optimization (MOPSO) is a highly efficient swarm intelligence optimization algorithm that has been widely applied to complex multi-objective decision-making problems such as engineering design, resource scheduling, and reservoir development. However, traditional MOPSO algorithms face severe challenges when dealing with high-dimensional, large-scale, or computationally expensive objective functions (such as reservoir numerical simulation). To obtain a high-quality Pareto optimal solution set, it is usually necessary to maintain a large population size, but this leads to a sharp increase in computational overhead for each generation, making the optimization process too time-consuming and failing to meet the timeliness requirements of practical engineering applications.

[0003] Existing research mainly focuses on two directions for improvement: first, reducing the population size during iteration to increase computational speed, but this easily leads to insufficient population diversity, causing the algorithm to get stuck in local optima and sacrificing the quality of the solution; second, introducing various diversity preservation mechanisms, but these often increase the complexity of the algorithm and fail to effectively solve the computational efficiency bottleneck. While some improved algorithms can reduce the population size, their deletion strategies lack adaptability. For example, they only perform a fixed percentage reduction in the early stages of iteration, failing to dynamically adjust according to the optimization process. This can easily lead to the accidental deletion of potentially valuable particles during the exploration phase or the retention of too many redundant particles during the development phase.

[0004] Existing patents and research largely focus on improving particle update rules, maintaining external archives, or combining them with other optimization ideas. They generally treat population size as a fixed parameter, failing to consider it as a dynamically adjustable core variable. In particular, there is a lack of an effective method to quantify particle value (contribution) and redundancy (crowding) and accordingly dynamically adjust the population size in stages and adaptively. To address these issues, this invention proposes a dynamic multi-objective particle swarm optimization method based on crowding and contribution, aiming to fill this technological gap. This method achieves synergistic optimization of computational efficiency and solution set quality through refined particle evaluation and dynamic pruning mechanisms. Summary of the Invention

[0005] To overcome the shortcomings of the existing methods, the purpose of this invention is to provide a dynamic population multi-objective particle swarm optimization method based on crowding and contribution. This method adaptively adjusts the population size according to different optimization problems and iteration processes, significantly improving computational efficiency while ensuring algorithm convergence and diversity. It provides an efficient and reliable tool for solving complex multi-objective engineering optimization problems with high computational costs.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] A dynamic population multi-objective particle swarm optimization method based on crowding and contribution is characterized by the following steps:

[0008] Step 1: Definition and initialization of core algorithm parameters:

[0009] 1.1 Basic parameter initialization:

[0010] Setting core algorithm parameters includes: initial population size Maximum number of iterations Inertia weight Learning factors (Individual cognitive factors) (Social cognitive factors), number of grid divisions Ideal number of particles per grid Deletion quantity limit ratio (Generally 10% of the total number of particles), iteration boundary point (Generally 50% of the total number of iterations).

[0011] 1.2 Particle Swarm Initialization

[0012] Based on the variable constraints of the optimization problem, an initial particle swarm is randomly generated. Each particle A candidate solution to the corresponding optimization problem includes 1 decision variable ( To optimize the problem's dimensionality, initialize the velocity of each particle. And calculate the objective function value for each particle. ( To optimize the number of targets.

[0013] Step 2: Solve spatial mesh generation

[0014] To achieve spatial localization and density assessment of particles in order to obtain the crowding degree and contribution of subsequent particles, the solution space is meshed. The specific steps are as follows:

[0015] (1) Each time the particle swarm loop is repeated, select all objective functions. maximum value and minimum value ;

[0016] (2) Define the range of values ​​for each objective function [ , Divided into equal parts Section, Section Chief After obtaining the maximum and minimum values ​​of all objective functions, each objective function It can be viewed as an edge, based on its and It can be divided into equal parts Segment, length of each segment This allows the resulting particle swarm to be divided into... In a small space, each particle can obtain the coordinates of its grid position.

[0017] Step 3: Calculation of Particle Contribution

[0018] Particle contribution is used to quantify the guiding value of a particle in the optimization search process, and it is calculated as follows:

[0019] The current population is ranked using non-dominated sorting and distance from the optimal solution, dividing particles into different positions. ( It refers to iteration At that time, particles The ranking of the particle; the reciprocal of this ranking is the particle's contribution ( The higher the rank, the greater the contribution, and the higher the guiding value of the particle for subsequent search directions.

[0020] Lower-ranked particles yield worse solutions than higher-ranked particles and may not bring significant benefits in subsequent searches, so this type of particle is highly likely to be deleted.

[0021] Step 4: Calculation of particle crowding degree

[0022] Particle crowding is a quantitative index of particle aggregation based on grid partitioning. This index reflects the distribution density of particles in the solution space: the higher the crowding, the more obvious the particle aggregation phenomenon within the grid, which is more likely to lead to similar solutions in the iterative search; therefore, particles in such grids are more likely to be deleted in subsequent iterations. The specific calculation formula is shown in the following formula (1):

[0023]

[0024] In the formula It refers to iteration At that time, particles The cell Particle crowding; It refers to iteration At that time, particles The cell The total number of particles.

[0025] Step 5, Dynamic Particle Removal Strategy:

[0026] Based on particle contribution With congestion Constructing particle deletion probability ( The process involves constructing a system and dynamically adjusting the deletion strategy during the iteration phase. The specific steps are as follows:

[0027] 5.1 Deletion probability calculation:

[0028] (1) Early stage of iteration ( In the early stage, the particles are relatively dispersed, and the deletion probability focuses on controlling the excessive aggregation of particles. The formula is shown in equation (2) below:

[0029]

[0030] (2) Later stage of iteration ( In the later stages, all particles tend towards the optimal solution, and their contributions all tend towards 1. If the earlier deletion formula is continued, the deletion probability will become 0. Therefore, the deletion probability is adjusted as follows: .

[0031] 5.2 Particle Filtering and Removal:

[0032] Generate random numbers in the interval [0,1]. For each particle, if > Then it will be included in the set of particles to be deleted. ;

[0033] Calculate the deletion limit, if the number of particles to be deleted is... Then select randomly. Delete one particle; if Then delete directly. All particles in it.

[0034] The beneficial effects of this invention are:

[0035] The method provided in this invention has the advantages of high computational efficiency and good solution set quality in solving multi-objective optimization problems. By introducing a dynamic pruning strategy and pruning restriction mechanism based on contribution and crowding, the method effectively balances the exploration and development capabilities of the algorithm. While significantly reducing computational costs, it ensures the convergence and uniformity of the obtained Pareto front. This method has important guiding and application value for solving computationally intensive engineering problems such as the co-optimization of CO2 flooding in oil reservoirs. Attached Figure Description

[0036] Figure 1 A flowchart illustrating the implementation of a dynamic population multi-objective particle swarm optimization algorithm based on crowding and contribution.

[0037] Figure 2The results are shown for different number of iterations. Detailed Implementation

[0038] The invention will be further described below with reference to the accompanying drawings.

[0039] To verify the performance superiority of the dynamic population multi-objective particle swarm optimization algorithm based on crowding and contribution, the standard multi-objective optimization test function ZDT1 was selected as the test object for comparative experiments. The ZDT1 function has the characteristics of being continuous, convex, and having a uniform distribution of Pareto optimal solutions, making it suitable for testing the convergence, distribution, and computational efficiency of multi-objective algorithms. It can accurately reflect the optimization capability of the proposed algorithm under dynamic population control.

[0040] During the experiment, such as Figure 1 As shown, firstly, according to the set parameter range and initialization rules, the solution space of the ZDT1 function is divided into a uniform grid of 81 cells. Then, for the initialized particle population, the contribution and crowding of each particle are calculated: the particle contribution is quantified based on its contribution to the Pareto optimal front, calculated by judging whether the particle is in the current optimal solution set and its contribution to improving the diversity of the solution set, thus selecting particles with core value for the optimization result; the particle crowding is achieved by calculating the distance between the particle and its neighboring particles in the target space, used to evaluate the uniformity of the population distribution and avoid particle aggregation that would limit the search range.

[0041] Based on the above calculation results, the dynamic particle deletion strategy designed in the implementation prioritizes the deletion of redundant particles with low contribution and high crowding, while retaining core particles with high contribution and uniform distribution. By dynamically adjusting the population size, the algorithm reduces unnecessary computation and improves its efficiency while ensuring optimization accuracy. During the experiments, variables were strictly controlled. Except for the population control strategy, the proposed algorithm maintained consistency with traditional MOPSO (Multi-Objective Particle Swarm Optimization) parameters such as inertia weight, learning factor, and maximum population size to ensure the fairness and effectiveness of the comparative experiments.

[0042] The calculation was performed for 10, 20, 40, and 80 iterations, with each iteration number run five times. The average value was used as the final result. The final calculation result and the time consumed are as follows: Figure 2 As shown in Tables 1 and 2:

[0043] Table 1. Timeline for different iteration numbers

[0044]

[0045] Table 2. Average Particle Contribution Parameters After 10 Iterations

[0046] The test function adopted is the ZDT1 test function, which is a widely used test function in the field of multi-objective optimization. It was proposed by Zitzler, Deb, Thiele, and others in 1999 and is mainly used to evaluate the performance of multi-objective evolutionary algorithms. As an evaluation function, the core of ZDT1 is to simulate the complexity and challenge of multi-objective optimization. Its formula is shown in the following equation (3):

[0047]

[0048] Algorithm parameter settings: Initial population size Maximum number of iterations Inertial weight Learning factor , Number of grid divisions Ideal number of particles per grid Deletion quantity limit ratio Iteration boundary point .

Claims

1. A dynamic population multi-objective particle swarm optimization method based on crowding and contribution, characterized in that, Includes the following steps: Step 1: Definition and initialization of core algorithm parameters: 1.1 Basic parameter initialization: Setting core algorithm parameters includes: initial population size Maximum number of iterations Inertia weight Learning factors (Individual cognitive factors) (Social cognitive factors), number of grid divisions Ideal number of particles per grid Deletion quantity limit ratio (Generally 10% of the total number of particles), iteration boundary point (Generally 50% of the total number of iterations); 1.2 Particle Swarm Initialization Based on the variable constraints of the optimization problem, an initial particle swarm is randomly generated. Each particle A candidate solution to the corresponding optimization problem includes 1 decision variable ( To optimize the dimensionality of the problem, initialize the velocity of each particle. And calculate the objective function value for each particle. ( To optimize the number of targets), step 2: Solve spatial mesh generation. To achieve spatial localization and density assessment of particles in order to obtain the crowding degree and contribution of subsequent particles, the solution space is meshed. The specific steps are as follows: (1) Each time the particle swarm loop is repeated, select all objective functions. maximum value and minimum value ; (2) Define the range of values ​​for each objective function [ , Divided into equal parts Section, Section Chief After obtaining the maximum and minimum values ​​of all objective functions, each objective function It can be viewed as an edge, based on its and It can be divided into equal parts Segment, length of each segment This allows the resulting particle swarm to be divided into... In a small space, each particle can obtain the coordinates of its grid position; Step 3: Calculation of Particle Contribution Particle contribution is used to quantify the guiding value of a particle in the optimization search process, and it is calculated as follows: The current population is ranked using non-dominated sorting and distance from the optimal solution, dividing particles into different positions. ( It refers to iteration At that time, particles The ranking of the particle; the reciprocal of this ranking is the particle's contribution ( The higher the ranking, the greater the contribution, indicating that the particle has a higher guiding value for subsequent search directions. Lower-ranked particles do not yield better solutions than higher-ranked particles and may not bring much benefit in subsequent searches, so this type of particle is highly likely to be deleted. Step 4: Calculation of particle crowding degree Particle crowding is a quantitative index of particle aggregation based on grid partitioning. This index is used to reflect the distribution density of particles in the solution space: the higher the crowding, the more obvious the particle aggregation phenomenon in the grid, which is more likely to lead to similar solutions in the iterative search; therefore, particles in such grids are more likely to be deleted in subsequent iterations. The specific calculation formula is shown in the following formula (1): ； In the formula It refers to iteration At that time, particles The cell Particle crowding; It refers to iteration At that time, particles The cell The total number of particles; Step 5, Dynamic Particle Removal Strategy: Based on particle contribution With congestion Constructing particle deletion probability ( The process involves constructing a system and dynamically adjusting the deletion strategy during the iteration phase. The specific steps are as follows: 5.1 Deletion probability calculation: (1) Early stage of iteration ( In the early stage, the particles are relatively dispersed, and the deletion probability focuses on controlling the excessive aggregation of particles. The formula is shown in equation (2) below: ； (2) Later stage of iteration ( In the later stages, all particles tend towards the optimal solution, and their contributions all tend towards 1. If the earlier deletion formula is continued, the deletion probability will become 0. Therefore, the deletion probability is adjusted as follows: ; 5.2 Particle Filtering and Removal: Generate random numbers in the interval [0,1]. For each particle, if > Then it will be included in the set of particles to be deleted. ; Calculate the deletion limit, if the number of particles to be deleted is... Then select randomly. Delete one particle; if Then delete directly. All particles in it.

Images