ACDB-EA: Adaptive convergence-diversity balanced evolutionary algorithm for many-objective optimization☆
Introduction
Many real-world applications have been described as many-objective optimization problems (MaOPs), such as engineering design [1], storage problem of blockchain [2], water allocation problem [3]. In this paper, we consider the following MaOP: where is called the decision space and indicates a candidate solution. is the objective space and consists of () objective functions to be minimized.
Many-objective evolutionary algorithms (MaOEAs) are population-based approaches [4], [5], [6], which aim to acquire a set of nondominated solutions achieving well convergence and diversity1 on the true Pareto front (PF) of MaOPs. During the last few decades, plenty of MaOEAs have been proposed and verified to be suitable for MaOPs [7], [8], [9], [10]. However, their ability of handling high-dimensional problems can be further explored, especially for Pareto-based MaOEAs. The main difficulty is that solutions become incomparable with each other in high-dimensional problems, which means the proportion of nondominated solutions increases. And such situation becomes worse with the increasing number of objectives, resulting in weaker selection pressure [11], [12]. The second is the difficulty of maintaining population diversity. Existing approaches for preserving the diversity may not be efficient in the high-dimensional objective space [13], such as th nearest distance [14] and crowding distance [15]. To tackle these two issues, researchers have developed many algorithms, and most of them are classified as three categories.
The first category is known as the Pareto dominance-based MaOEAs, which enhances the selection pressure by modifying the definition of classical Pareto dominance, making solutions be able to compare with each other. For instance, fuzzy dominance [16], -dominance [17], -dominance [18] adopt modified Pareto dominance to choose solutions. Tian et al. [19] designed a novel dominance relation to obtain better balance between convergence and diversity based on an adaptive technique. Qiu et al. [20] proposed fractional dominance to strengthen convergence by comparing the value of each objective between two solutions. Besides, another method is to combine the classical dominance and an additional convergence-related metric. Zhang et al. [21] proposed knee point driven evolutionary algorithm (KnEA) to enhance the convergence performance.
The second category covers the indicator-based MaOEAs, which adopts indicators as the selection criterion to assess nondominated solutions and guide the evolutionary procedure. The R2 indicator [22], the S-metric [23], indicator [24], and the hypervolume (HV) indicator [25] are widely used indicators. For example, HV-based MaOEAs [26], [27], [28] evaluate solutions according to their contributions to the HV value. Moreover, arbitrary indicators can be integrated into the indicator-based evolutionary algorithm (IBEA) [24].
The third category represents the decomposition-based MaOEAs, which decomposes a complex MaOP into simpler multiobjective subproblems [29] or a group of single-objective optimization problems (SOPs) [30], and then solve them collaboratively. Two representative algorithms are MOEA/D [30] and NSGA-III [13]. In MOEA/D, each subproblem is optimized based on the information on its several neighboring subproblems, which enables to reduce computational complexity. In NSGA-III, a set of reference points are adopted to decompose the objective space, which promotes the diversity. Following the basic idea of them, some environmental selection strategies have been proposed. Li et al. [31] combined dominance and decomposition, while Yuan et al. [32] exploited the perpendicular distance from solutions to reference vectors in the objective space. To tackle the issue of lacking the information of true PFs beforehand, some effective approaches are proposed, such as adapting the weights during the evolutionary process [33], learning the distribution of reference vectors by the growing neural gas (GNG) network [34] and extracting reference vectors by a modified -means clustering method [35].
There are also some MaOEAs that are not located in the above categories, such as determinantal point process-based algorithm [36], adaptive clustering-based algorithm [37], two-archive algorithm [38], [39] (i.e., convergence archive and diversity archive), dimension reduction and solving knowledge guided algorithm [40], voting-based algorithm [41], two-stage algorithm [42] and decision variable classification-based algorithm [43]. Ding et al. [41] proposed an effective multi-stage evolutionary strategy based on knowledge fusion. The statistical guidance vector was designed as one of the knowledge, which calculated the concentration degree of elite solutions on the basis of voting method. Li et al. [42] solved the objective optimization by a convergence stage and a diversity stage. In the convergence stage, the individuals competed with each other. In the diversity stage, the solutions with well space-filling were chosen. Liu et al. [43] developed a novel decision variable classification method to generate promising solutions, which achieved a higher probability to produce offspring with well convergence and diversity.
Although there are a lot of research on MaOEAs, handling the situation to balance convergence and diversity in high-dimensional problems still needs to be further explored [44], [45], [46]. Algorithms need to promote good diversity of the population, while guarantee high selection pressure upon the true PFs. The maximal marginal relevance (MMR) [47] is a popular approach in improving diversity, which is widely used in text retrieval [48], document summarization [47], product recommendation [49], and so on. The way of generating document summarization gives us a motivation of this paper. Further study is explored based on the above work. Specifically, when choosing solutions into the next generation, the solution which achieves well convergence towards the true PFs and contains the minimal similarity to the previously selected solutions should be chosen firstly. This way, the performance of balancing convergence and diversity can be improved to obtain satisfactory final population.
Inspired by the work in [47], we propose an adaptive convergence-diversity balanced evolutionary algorithm (ACDB-EA) to solve the problem of maintaining the balance between convergence and diversity. In the proposed algorithm, the contribution of each solution is quantitatively calculated by implementing the adaptive convergence-diversity balanced strategy, which produces a score based on convergence, global and local diversities. Solution with the highest contribution will be selected into the next generation, which means it obtains the better balance between convergence and diversity. To be specific, the convergence of a solution is defined as the norm in the objective space, and the diversity of a solution is measured by its similarity in the population, where the similarity between two solutions is defined as the cosine similarity between their objective vectors. In this way, the average similarity and the maximal similarity represent respectively the global and local diversities of the solution, aiming to enhance diversity. The adaptive convergence-diversity balanced strategy adjusts weights of convergence and diversity, global diversity and local diversity according to the population adaptively, which aims to make an appropriate tradeoff between convergence and diversity. To promote diversity in the objective space, a clustering strategy is also adopted in this paper. Furthermore, to enhance the quality of the mating pool, we develop a convergence-crowding-based mating strategy and an elitist solution archive. In the convergence-crowding-based mating strategy, selection of the mating parent depends on the quality of convergence and crowding. The harmonic average distance (HAD) [50] is adopted to reflect the crowding status of solutions in the objective space, which estimates the density around one solution by using the -nearest neighbor distances. In elitist solution archive, elitist solutions with good convergence and diversity on each objective dimension are preserved on each generation.
The main contributions of this paper are summarized as follows.
- (1)
A convergence-crowding-based mating strategy which selects the mating parent based on convergence and crowding is proposed to balance convergence and diversity in the mating pool. To be specific, the norm and the HAD method are used to reflect the convergence and crowding status of solutions, respectively. This strategy gives corresponding probabilities and of choosing the mating parent based on the convergence and crowding.
- (2)
An elitist solution archive, termed ESA, is suggested to preserve elitist solutions with good convergence and diversity on each objective dimension. In the process of evolution, ESA updates at every generation to promote the mating pool, aiming to produce high-quality offsprings.
- (3)
An adaptive convergence-diversity balanced (ACDB) strategy is developed to quantitatively calculate the score of each solution based on the convergence, global diversity and local diversity. Solution with the highest score obtains the better balance between convergence and diversity. By selecting the solution with the highest score in each iteration, the proposed algorithm achieves well balance between convergence and diversity.
The remainder of this paper is shown as follows. The related work is summarized in Section 2. The preliminaries are described in Section 3. Section 4 elaborates the proposed ACDB-EA. Experimental results are presented and discussed in Section 5. In the end, the conclusions and future work are drawn in Section 6.
Section snippets
MaOEAs in balancing convergence and diversity
There are many MaOEAs have been proposed based on the idea of balancing the convergence and diversity. Methods such as modifying the dominance relation [19], using cosine similarity [36], [51], performing decomposition operation in the objective space [33], [44], [52], performing the selection operation depending on the value of the angle between the vectors [13], [53], clustering in the objective and decision spaces [37], [54], [55] have been suggested in many studies recently. In [19], a
Diversity maintenance mechanism based on the global and local diversities
As introduced above, existing diversity maintenance mechanisms include th nearest distance-based mechanism [14], crowding distance-based mechanism [15], reference vector-based mechanism [30], and so on. In the crowding distance-based mechanism, the density of solutions surrounding a certain solution is estimated to measure the crowding status of the solution, which reflects the diversity of the solution to some extent. This kind of mechanism focuses more on the local diversity of the solution,
General framework of ACDB-EA
The pseudocode of ACDB-EA is presented in Algorithm 1. At first, initialize population with solutions randomly (line 1). Then, of size is derived from (line 2). Afterwards, the ideal point and the nadir point are identified by and represent the minimum and the maximum values on each objective , , respectively (line 3). To alleviate the impact of different scaled objectives in MaOPs, the
Experiments
The proposed ACDB-EA is evaluated and analyzed in this section. Specifically, we carry out experiments on 111 benchmark testing instances with 2–20 objectives optimization problems and compare the proposed method with seven state-of-the-art MaOEAs: hpaEA [72], NSGA-II/SDR [19], DEA-GNG [69], AdaW [33], MOEA/D-UR [73], MultiGPO [71] and PeEA [70]. Experiments are carried out on a PC with Intel(R) Xeon(R) E5-2683 v4 CPU @ 2.10 GHz and 128 GB RAM. All compared algorithms and ACDB-EA are executed
Conclusion
In this paper, an adaptive convergence-diversity balanced evolutionary algorithm is proposed, termed ACDB-EA, which aims to maintain well balance between convergence and diversity of the obtained solution set during the evolutionary process. In the proposed approach, an adaptive convergence-diversity balanced (ACDB) strategy is introduced to adjust the weights of convergence and diversity adaptively, where a diversity maintenance mechanism based on the global and local diversities is adopted.
CRediT authorship contribution statement
Yu Zhou: Conceptualization, Investigation, Methodology, Software, Formal analysis, Writing – original draft. Sheng Li: Formal analysis, Writing – original draft. Witold Pedrycz: Check. Guorui Feng: Conceptualization, Funding acquisition, Resources, Supervision.
Declaration of Competing Interest
No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.swevo.2022.101145.
References (85)
- et al.
Multiobjective evolutionary algorithms: A survey of the state of the art
Swarm Evol. Comput.
(2011) - et al.
Differential evolution using improved crowding distance for multimodal multiobjective optimization
Swarm Evol. Comput.
(2021) - et al.
Applying graph-based differential grouping for multiobjective large-scale optimization
Swarm Evol. Comput.
(2020) - et al.
Improved NSGA-III with selection-and-elimination operator
Swarm Evol. Comput.
(2019) - et al.
One-to-one ensemble mechanism for decomposition-based multi-objective optimization
Swarm Evol. Comput.
(2022) - et al.
Evolutionary many-objective algorithm based on fractional dominance relation and improved objective space decomposition strategy
Swarm Evol. Comput.
(2021) - et al.
SMS-EMOA: Multiobjective selection based on dominated hypervolume
European J. Oper. Res.
(2007) - et al.
A multi-stage knowledge-guided evolutionary algorithm for large-scale sparse multi-objective optimization problems
Swarm Evol. Comput.
(2022) - et al.
A two-stage surrogate-assisted evolutionary algorithm (TS-SAEA) for expensive multi/many-objective optimization
Swarm Evol. Comput.
(2022) - et al.
A multiobjective evolutionary algorithm based on decision variable classification for many-objective optimization
Swarm Evol. Comput.
(2022)
DCDG-EA: Dynamic convergence–diversity guided evolutionary algorithm for many-objective optimization
Expert Syst. Appl.
A many-objective evolutionary algorithm with diversity-first based environmental selection
Swarm Evol. Comput.
On the estimation of Pareto front and dimensional similarity in many-objective evolutionary algorithm
Inform. Sci.
A decomposition-based many-objective evolutionary algorithm updating weights when required
Swarm Evol. Comput.
Comparison between MOEA/D and NSGA-III on a set of novel many and multi-objective benchmark problems with challenging difficulties
Swarm Evol. Comput.
Many-objective optimization: An engineering design perspective
On cloud storage optimization of blockchain with a clustering-based genetic algorithm
IEEE Internet Things J.
Managing population and drought risks using many-objective water portfolio planning under uncertainty
Water Resour. Res.
Investigating the properties of indicators and an evolutionary many-objective algorithm Using Promising Regions
IEEE Trans. Evol. Comput.
Evolutionary multi-criterion optimization
On finding pareto-optimal solutions through dimensionality reduction for certain large-dimensional multi-objective optimization problems
Kangal Rep.
On the evolutionary optimization of many conflicting objectives
IEEE Trans. Evol. Comput.
An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, Part I: solving problems with box constraints
IEEE Trans. Evol. Comput.
A fast and elitist multiobjective genetic algorithm: NSGA-II
IEEE Trans. Evol. Comput.
Fuzzy-based Pareto optimality for many-objective evolutionary algorithms
IEEE Trans. Evol. Comput.
Borg: An auto-adaptive many-objective evolutionary computing framework
Evol. Comput.
A new dominance relation-based evolutionary algorithm for many-objective optimization
IEEE Trans. Evol. Comput.
A strengthened dominance relation considering convergence and diversity for evolutionary many-objective optimization
IEEE Trans. Evol. Comput.
A knee point-driven evolutionary algorithm for many-objective optimization
IEEE Trans. Evol. Comput.
Indicator-based selection in multiobjective search
A faster algorithm for calculating hypervolume
IEEE Trans. Evol. Comput.
Improving hypervolume-based multiobjective evolutionary algorithms by using objective reduction methods
An EMO algorithm using the hypervolume measure as selection criterion
Covariance matrix adaptation for multi-objective optimization
Evol. Comput.
Decomposition of a multiobjective optimization problem into a number of simple multiobjective subproblems
IEEE Trans. Evol. Comput.
MOEA/D: A multiobjective evolutionary algorithm based on decomposition
IEEE Trans. Evol. Comput.
An evolutionary many-objective optimization algorithm based on dominance and decomposition
IEEE Trans. Evol. Comput.
Balancing convergence and diversity in decomposition-based many-objective optimizers
IEEE Trans. Evol. Comput.
What weights work for you? Adapting weights for any Pareto front shape in decomposition-based evolutionary multiobjective optimisation
Evol. Comput.
An Adaptive Reference Vector Guided Evolutionary Algorithm Using Growing Neural Gas for Many-Objective Optimization of Irregular ProblemsTech. Rep.
Cited by (14)
A many-objective evolutionary algorithm under diversity-first selection based framework
2024, Expert Systems with ApplicationsAn objective reduction algorithm based on population decomposition and hyperplane approximation
2024, Swarm and Evolutionary ComputationAn enhanced Equilibrium Optimizer for solving complex optimization problems
2024, Information SciencesA dual distance dominance based evolutionary algorithm with selection-replacement operator for many-objective optimization
2024, Expert Systems with ApplicationsA constrained multi-objective evolutionary algorithm with two-stage resources allocation
2023, Swarm and Evolutionary ComputationA strength pareto evolutionary algorithm based on adaptive reference points for solving irregular fronts
2023, Information Sciences
- ☆
This work was supported by the Science and Technology Planning Project of Zhejiang Province (2022C01090) and the National Natural Science Foundation of China under Grants 62072295 and 62072114.