Reliability of the weight vector generation method of the multi-objective evolutionary algorithm and application

Highlights

• This paper proposes a multi-stage and multi-objective algorithm.

• A method of judging the reliability of the weight vector is proposed.

• Proposed a way to find high-quality solutions.

• Compare with other 15 algorithms in 47 test questions.

• Optimized mechanical bearings and gear reducers.

Abstract

The decomposition-based multi-objective evolutionary algorithm first generates a set of weight vectors in advance, and it is very important to select a set of appropriate weight vectors for the decomposition-based algorithm. A variety of weight vector generation methods have been proposed in the existing algorithms, but in most algorithms, a pre-defined weight vector generation method is still used, the pre-defined weight vector is too specialized for the simplex-like front surface, which results in poor performance on the front surface with irregularities. At the same time, most of the existing algorithms have proposed many new adaptive strategies for weight vectors, but if you generate a set of more suitable weight vectors at the beginning, and then use the update strategy, it can make the algorithm achieve a better balance between diversity and convergence. In order to select a suitable weight vector, this paper proposes a multi-stage MOEA to select a suitable weight vector. The algorithm is divided into multiple stages according to the evolution process, first of all, in the early stage of evolution, the reliability of multiple weight vector generation methods was evaluated according to the spearman correlation coefficient in statistics, choose the most suitable weight generation method; Secondly, this method can be applied to the search for high-quality solutions in the middle of evolution; Finally, a weight vector adaptive strategy is adopted in the overall evolution process. In the experiment, the proposed algorithm was analyzed in the benchmark test problem, mechanical bearing and light aircraft gear reducer. The experimental results show the effectiveness of the proposed algorithm.

Introduction

Realistic multi-objective optimization problems are composed of two or more simultaneous optimization objectives [1], at the same time, multiple problems are in conflict with each other, and the optimal solution of one optimization goal is not necessarily the optimal solution of the other optimization goals. This means that it is not possible to find a single solution that optimizes all goals at the same time. For example, in the optimization of rolling bearings, it is necessary to achieve the dynamic capacity and the minimum oil film thickness at the same time to achieve the maximum [2] [3], but in this problem, as the dynamic capacity increases, the minimum oil film thickness is gradually decreasing. There is no single solution that can maximize the dynamic capacity and the minimum oil film thickness at the same time. The ultimate solution to this problem is to find a relatively optimal boundary curve, which is to find the Pareto frontier of the optimization target, find a group among them that can guarantee a higher oil film thickness even at a higher dynamic capacity. So to solve a multi-objective problem is to find a trade-off solution among multiple objectives, this solution is called the Pareto optimal solution, and the front surface formed by all the solutions is called the Pareto frontier [4].

In the current existing multi-objective evolutionary algorithms, it can be roughly divided into three directions. In the framework based on Pareto dominance, the Pareto dominance principle plays a key role in this framework, for example, the most classic non-dominated sorting genetic algorithm NSGA-II [5]. In this direction, the main focus is to solve the Pareto dominance principle. When solving high-dimensional problems, the proportion of non-dominated solutions in the population will increase exponentially as the dimensionality increases, leading to a serious loss of selection pressure on the frontier of the population [6] [7], in the existing research, there are many ways to solve the insufficiency of the Pareto dominance principle in the high-dimensional problem, for example, average and maximum ranking [8], preference relationship [9], fuzzy Pareto dominance [10], extended relationship [11], etc.

In an indicator-based framework, AR-MOEA is a representative indicator-based algorithm [12], it is an algorithm based on greedy strategy and reference point adaptation. In the indicator-based direction, it is widely used due to the establishment of the hypervolume indicator theory. Many research works are designed around the predefined binary indicator, the hypervolume indicator, GD and R2 indicator [13], [14], [15], [16].

In the decomposition-based framework, the decomposition-based idea has been applied to a certain degree in several meta-heuristic algorithms [17], [18], [19], [20], [21], [22], [23], [24], the multi-objective framework based on decomposition finally proposed the MOEA/D algorithm in 2007 [25]. The principle of decomposition is to decompose a multi-objective problem into multiple single-objective or multiple simple multi-objective problems for optimization. The focus of this article is also to improve the algorithm in the decomposition-based framework.

In the decomposition process, it is very important to have a set of appropriate weight vectors, which can make full use of computing resources in the calculation process, and at the same time achieve a good balance between convergence and diversity. Among the existing decomposition-based algorithms, many algorithms propose a new weight vector generation method, or propose a weight vector adaptive strategy to make the weight vector update according to the offspring. The innovation based on the decomposition algorithm is mainly reflected in the generation method of the weight vector, the decomposition method, the calculation resource allocation strategy, the mating selection mechanism, the duplication operator and the replacement process. For example, the MOEA/D-AWA algorithm is proposed in Literature [26], when the target problem has a complex Pareto, a uniformly distributed weight vector can only guarantee the diversity of the optimal solution, and a strategy for adaptive adjustment of the weight vector is proposed. By calculating the crowding distance between the populations to add and delete operations, the weight vector is automatically adjusted periodically, which enhances the versatility of the algorithm in the target problem. The MOEA/D-CMA algorithm is proposed in the literature [27], it is difficult for multi-objective algorithms to solve test problems with bias characteristics. The adaptive evolution strategy and differential evolution strategy using the covariance matrix with good performance in the search space are proposed, through experimental comparison, it is shown that the proposed algorithm has better performance in dealing with problems with deviations. The MOEA/D-D algorithm is proposed in the literature [28]. The key problem of multi-objective optimization is to achieve a good balance between population convergence and diversity. This algorithm combines Pareto dominance and decomposition-based methods, and uses the advantages of both to evaluate convergence and diversity in the evolutionary process, at the same time, the algorithm has played a very good performance in many objective problems, and has shown good performance in constrained optimization problems. The MOEA/D-MRDL algorithm is proposed in [29]. The paper proposes that the existing diversity measurement methods are all performed in an offline form. Aiming at most algorithms that require accurate Pareto optimal frontiers and ideal vectors, an online method of measuring diversity is proposed, find out the contribution of any individual in the current population to the performance of diversity, and determine the retention strategy of the individual according to the size of the contribution, make the algorithm have better performance in diversity. Literature [30] proposed the MOEA/D-PaS algorithm. The paper analyzes the commonly used scalarization methods and shows that there is a method to maximize the search ability based on the decomposition algorithm, and given a set of pre-weighted vectors, any solution along the PF can be found. Therefore, an adaptive scalar approximation strategy is proposed to stably achieve the maximum search capability of the decomposition algorithm. Literature [31] proposed the MOEA/D-STM algorithm, and proposed a simple and effective matching mechanism to coordinate the selection process of the population. Matching the sub-problems with a single solution weighs the balance between diversity and convergence in the evolutionary search process. Literature [32] proposed the MOEA/D-FRRMAB algorithm, and proposed an adaptive operator selection method, according to the performance of different operators in the optimization process, the optimal operator is selected online. Literature [33] proposed the MOEA/D-M2M algorithm, in which it is suggested to solve the sub-problems in a collaborative way during the evolution process, each different sub-problem has its own population, which is similar to splitting a multi-objective optimization problem into multiple simple multi-objective optimization problems. This Literature points out that through this split method, the diversity of the population can be well maintained. Literature [34] proposed the MOEA/D-DU algorithm, which pointed out that due to the nature of the contour of the aggregate function, it is usually unable to maintain good diversity in high-dimensional targets. In order to solve this problem, a solution to the vertical distance of the weight vector in the target space is proposed to effectively maintain the balance between diversity and convergence, according to the experimental results, this method is very competitive. Literature [35] proposed the MOEA/D-URAW algorithm. The paper proposes that a decomposition-based multi-objective evolutionary algorithm using appropriate weights may obtain a higher quality final solution set. Proposed a random weight generation method and weight adaptation based on sparsity to obtain a better weight distribution. Finally, tests were carried out in a variety of test questions to verify the effectiveness of this method. As shown in the Table 1, this paper gives a table of the above algorithms, the first column is the abbreviation of the algorithm and the second column is the full name of the algorithm.

Obtained from the above documents, in many decomposition-based algorithms, uniformly distributed weight vectors are generated in advance. In the MOEA/D-AWA algorithm and the MOEA/D-URAW algorithm, two methods of generating weight vectors are respectively proposed, namely the weight obtained by the WS transform and the uniformly randomly distributed weight vector. There are also hybrid weight vector generation methods [36] and incremental grid weight vector generation methods [37]. Most algorithms still use a pre-defined weight vector generation method, the pre-defined weight vector is too specialized for the simplex-like front surface, this results in poor performance on the front surface with irregularities. Algorithms based on decomposition must have a set of weight vectors to achieve the purpose of decomposition. Therefore, the method of generating weight vectors is different, which leads to the degree of decomposition, even though a variety of weight vector generation methods are proposed, the performance of different generation methods in different problems is quite different. Among the existing algorithms, few have proposed a multi-stage algorithm to solve the problem of how to select the appropriate weight vector. This paper proposes to use the Spearman correlation coefficient in statistics to evaluate the reliability of multiple weight vector generation methods, to judge the reliability of different weight vectors in different problems, each time the weight vector generation method with the highest reliability is selected as the weight in the evolution process. In addition, this article uses the spearman correlation coefficient in statistics to evaluate the reliability because it has been well used in many research backgrounds [38], [39], [40], [41], [42], and the method discussion and experimental comparison below have confirmed the feasibility of this method. The Spearman correlation coefficient can also be used in the evaluation of high-quality solutions, which is similar to the non-dominated sorting of the solution set. Experimental verification shows the effectiveness of the method.

To sum up, a multi-stage decomposition-based multi-objective evolutionary algorithm MOEA/D-R is proposed in this paper, and a reliability evaluation method of weight vector and high-quality solution is proposed. This evaluation method uses the Spearman correlation coefficient method to judge the reliability of various weight vectors. In this paper, the weight vector generation method with high reliability refers to the generation method that is more suitable for the target problem among the multiple weight vector generation methods, that is, the weight vector with the highest Spearman correlation. A high-quality solution with high reliability refers to a solution set that is more in line with the evolutionary direction in the offspring population when generating offspring in the population, and is put into the external population to guide the evolution of the population. The main contributions of this article are as follows:

• In view of the fact that there are fewer algorithms in the above existing algorithms that evaluate the reliability of the weight vector and the solution set, we propose a multi-stage multi-objective algorithm, called as the multi-objective algorithm based on reliability evaluation (MOEA/D-R), the evolution process is divided into two stages, and the most reliable weights and high-quality solutions are selected in different stages. This method solves the problem that the pre-defined weight vector in the decomposition algorithm is too specialized for the simplex-like front surface, resulting in poor performance on the irregular front surface. The algorithm first analyzes the reliability of the pre-generated weight vector, and selects a generation method with the highest reliability.

• The proposed reliability evaluation method can replace the principle of non-domination to find a better solution set in the process of population evolution. In the middle and late stages of population evolution, use this method to evaluate the high-quality solutions found in the search process, the use of high-quality solutions has good convergence and diversity characteristics to guide the population to evolve in the optimal direction.

• In order to effectively prove the effectiveness of our proposed strategy, in the experiment, we conducted experiments on 47 kinds of benchmark test problems and two real-world problems optimization, two practical problems are mechanical bearing problems and light aircraft gear reducers. 47 types of test problems are also classified into low complexity frontiers, more complex frontiers and highly complex frontiers. The performance of the proposed algorithm is verified, and it is suitable for real problems. By comparing with other 15 different algorithms, the experimental results show that MOEA/D-R has good competitiveness.

The rest of this article is organized as follows. The second section introduces the basic knowledge of multi-objective optimization, it explains the four weight vector generation methods used in this article and the offspring update strategy used in this article, which is similar to other documents. The third section expounds the proposed algorithm, from the basic idea to the whole process of algorithm realization. In the fourth section, experiments and discussions were carried out in 47 benchmark test questions. In the fifth section, the mechanical bearings and light aircraft gear reducers are optimized, and compared with other algorithms, experiments and discussions are carried out. Finally, in the sixth section, a summary and prospects for future research are given.