In practical engineering, complex multi-objective problems often lead to losses in the convergence and diversity of optimization results. To address this issue, an auxiliary point evolutionary algorithm based on KnEA (A Knee Point Driven Evolutionary Algorithm) is proposed in this paper. This algorithm uses inflection points with obvious trends in the evolution process as the centers of neighborhood spaces. By analyzing the relationship between inflection points in the neighborhood space and those of the previous generation, it quantitatively assesses the potential of the neighborhood space. Additionally, to overcome the convergence deficiency in optimizing complex problems, a strategy of adding auxiliary points based on the evolution potential of the neighborhood space is introduced, enhancing the spatial search ability during evolution. In the environmental selection of the population, inflection points and auxiliary points are preferentially selected for the next generation to balance convergence and diversity. Experimental results on multi-objective test problems demonstrate the significant advantages of the proposed algorithm. Finally, the algorithm is applied to the optimization design of a two-stage helicical cylindrical gear reducer.
1. Introduction
Multi-objective optimization problems have become crucial in various fields such as engineering, biology, and ecology. In recent years, these problems have shown an increasingly complex trend in form, imposing stricter requirements on the optimization performance of multi-objective optimization algorithms in practical applications. During the multi-objective optimization process, excessive emphasis on maintaining diversity can lead to insufficient convergence of solutions, while 过度追求收敛性会导致解的多样性丢失。
To address these challenges, diverse strategies have been adopted in the population evolution process. One approach is the multi-subpopulation strategy, with grid partitioning being a notable example. For instance, the literature [3] utilized grid partitioning-based memetic clustering analysis to enhance population diversity. However, determining appropriate parameter settings for grid strategies remains a challenge. Another approach involves adaptively selecting individuals with obvious trends on the Pareto front to adjust the diversity and convergence of the population. The literature [4] proposed a method of gradually selecting individuals with optimal convergence in environmental selection for the next generation, thereby improving both population convergence and diversity. The literature [5] suggested using inflection points with obvious trends as the preferred individuals in environmental selection and adaptively partitioning the neighborhood range of inflection points to increase the convergence rate while maintaining diversity.
Despite the ability to enhance population selection pressure through individuals with obvious advantages, the problem of insufficient convergence persists when dealing with complex problems. Therefore, to better maintain population diversity and improve convergence in solving complex multi-objective problems, this paper proposes an auxiliary point evolutionary algorithm AKnEA (KnEA-Based Assistant Point Evolution Algorithm) based on KnEA. AKnEA divides the target space into multiple subspaces by establishing a spatial environment centered around each inflection point. It assesses the evolution potential of the target space by analyzing the evolutionary distance between consecutive generations of inflection points. To boost the competitive ability of the potential space during evolution, neighboring points of the inflection points are additionally selected for the next generation to enhance the exploration ability within the current space, gradually guiding the population towards the true Pareto front. To validate the performance of the AKnEA algorithm, it is compared with three popular evolutionary algorithms on the WFG series of test functions. Finally, to demonstrate the practical applicability of the algorithm, it is applied to the optimization design of a two-stage helicical cylindrical gear reducer.
2. KnEA Principle
In KnEA, inflection points are identified by the distance from non-dominated solutions to the extreme points forming a hyperplane, and an adaptive strategy is employed to adjust the neighborhood size of each inflection point.
3. Auxiliary Point Evolutionary Algorithm Based on KnEA
3.1 Evolutionary Potential of Neighborhood Space
To provide more effective guidance for the population evolution direction, a neighborhood potential strategy is proposed. The potential of the neighborhood space is analyzed based on the neighborhood space where the inflection point is located, as shown in Figure 2.
The shaded region in the figure represents the neighborhood space centered around inflection point C. The size of the neighborhood space is expressed by the cosine similarity between the individual near the neighborhood boundary and the inflection point C. The formulas for the upper and lower bounds of the neighborhood space size are as follows:
where and represent the upper and lower bounds of the neighborhood space, respectively.
To analyze the potential value of the current inflection point’s neighborhood space, it is explored whether there are inflection points from the previous generation within this space. There are two cases: one is that no previous-generation inflection point is found in the neighborhood space, and the other is that a previous-generation inflection point is detected. When there is exactly one previous-generation inflection point in the neighborhood space, the Euclidean distance d between and the current inflection point C is calculated to quantitatively analyze the potential of this neighborhood space.
3.2 Auxiliary Point Strategy
When dealing with complex multi-objective problems, the population may experience excessive diversity and insufficient convergence during evolution. Therefore, enhancing the search intensity in the potential space of the target space can further improve the convergence of the evolutionary algorithm.
After evaluating the potential of the inflection point’s neighborhood space, auxiliary points are adaptively added to the potential space based on the evaluation result to enhance the search ability. The relationship between the average Euclidean distance Hm between the individuals in the neighborhood and the inflection point and the distance d is used as the basis for determining whether to add auxiliary points to the space. The formula for calculating the average distance Hm is as follows:
where i represents the index of the inflection point, and represents the Euclidean distance from the j-th inflection point to the inflection point in the previous generation within the neighborhood. When , a random individual within the neighborhood is selected as an auxiliary point for the next generation.
3.3 Algorithm Flow
The flowchart of the proposed AKnEA algorithm is presented in Figure 3. The algorithm begins by initializing the population and identifying the inflection points. Then, it evaluates the evolutionary potential of the neighborhood space of each inflection point and determines whether to add auxiliary points based on the potential. After that, the non-dominated solutions are selected, and the population is updated. This process is repeated until the termination condition is met.
4. Simulation Testing of AKnEA
4.1 Test Functions and Parameter Settings
The performance of the AKnEA algorithm is verified using the WFG series of test functions. The proposed algorithm is compared with MOEAD, MMOPSO, and KnEA on the WFG series test functions with 3 and 4 objectives. The parameter settings of the compared algorithms are adopted from the original literature. In the algorithm’s solution process for the test functions, the evaluation times are set to 10,000, and the number of individuals in the population is set to 100. All tests are independently run 30 times, and the average and standard deviation of the performance index values are recorded. Additionally, the Wilcoxon rank-sum test with a significance level of 0.05 is used for statistical analysis of the experimental results. The symbols “+”, “-“, and “≈” indicate that the algorithm’s index result is significantly better, significantly worse, and statistically similar to that of the AKnEA algorithm, respectively.
4.2 Algorithm Performance Evaluation Index
The performance of the algorithm is evaluated using the Inverted Generational Distance (IGD). For each test problem, approximately 5,000 points are uniformly sampled on the true Pareto front to calculate the IGD value. A smaller IGD value indicates better algorithm performance.
4.3 Test Results Analysis
Table 1 shows the average IGD values, variances, and rank-sum test results of the MOEAD, MMOPSO, KnEA, and AKnEA algorithms on the 3-objective WFG4-7 test functions. The shaded part indicates that the IGD values of the proposed algorithm are smaller than those of the other compared algorithms. From the rank-sum test results, it can be seen that the performance of AKnEA is significantly better than that of the other algorithms. Compared with KnEA, the standard deviation of the proposed algorithm is significantly smaller on the 3-objective WFG7 test function, indicating that the proposed algorithm also has an advantage in the stability of the optimization effect.
To further examine the performance of the algorithm on high-dimensional complex problems, the proposed algorithm is tested on the 4-objective WFG4-7 test functions. The test results of the four algorithms are shown in Table 2. The experimental results once again verify the superiority of AKnEA. More specifically, the optimization effects of MOEAD and MMOPSO on the 4-objective problems are significantly worse than that of the proposed algorithm. In addition, compared with KnEA, the standard deviation of the optimization results of the proposed algorithm is smaller.
To better illustrate the performance of AKnEA, the Pareto fronts of the AKnEA algorithm on the 3-objective and 4-objective WFG6 test problems are shown in Figure 4. It can be seen from the figure that the proposed algorithm performs well in maintaining both population diversity and convergence.
5. Engineering Example
5.1 Multi-Objective Optimization Model of the Reducer
5.1.1 Design Variables
The design variables of the two-stage helical cylindrical gear reducer are as follows:
where and are the normal module of the two gears, and are the number of teeth of the high-speed and low-speed stage pinions of the reducer, respectively, is the transmission ratio of the high-speed stage of the reducer, and are the helix angles of the reference circle, and and are the face width coefficients.
The number of teeth of the high-speed stage gear is calculated as , and the number of teeth of the low-speed stage gear is calculated as .
5.1.2 Objective Functions
To reduce the production cost of the reducer, minimizing the center distance and gear thickness of the reducer are the two optimization objectives. The objective functions are as follows。
5.2 Reducer Optimization Design and Comparative Analysis
In the design of a belt conveyor drive, the power transmitted by the reducer is 4.15 kW, the maximum input speed is [input speed value], the output speed is [output speed value], and the total transmission ratio . The AKnEA algorithm is used to perform multi-objective optimization design on the reducer. The population size is set to 200, and the evaluation times are set to 50,000.
