A point crowding-degree based evolutionary algorithm for many-objective optimization

It is well known that many-objective optimization problems (MaOPs) are difficultly to be balanced diversity and convergence in the search process because diversity and convergence contradict each other. The method of evaluating the diversity of the solution set can directly affect the final performances of the algorithms. In this article, a point crowding-degree (PC) strategy is proposed to evaluate the diversity of solutions. The proposed PC strategy not only considers the distance between any two points as large as possible, but also considers the gap between each dimension component as large as possible. Moreover, The PC only ponders the influence of surrounding neighbor points on the diversity of a point. A selection strategy is designed to balance convergence and diversity. Based on all, a point crowding-degree based evolutionary algorithm (PCEA) for many-objective optimization problems is proposed. The PCEA is compared experimentally with several state-of-the-art algorithms on the 57 many-objective benchmark functions and the experimental results show that the proposed PCEA algorithm has strong competitiveness and better overall performance. In addition, the proposed PC strategy is integrated into other advanced MaOPs methods. The results show that it is beneficial to improve the performance of other MaOEAs algorithms.


Introduction
Evolutionary algorithms and memetic algorithms have been widely used in dealing with various problems [1][2][3][4][5][6], one of the important applications is the multi-objective optimization problem (MOPs).Generally speaking, a minimization MOP is defined as: min F(X ) = ( f 1 (X ), f 2 (X ), . . ., f M (X )), subject to X R D (1) where X = (x 1 , x 2 , . . ., x D ) is a solution vector in the decision space of D dimensions (i.e., variables) and F = [ f 1 , f 2 . . ., f M ] is an objective vector in the objective space B Cai Dai cdai0320@snnu.edu.cnCheng Peng 2285112602@qq.comXiujuan Lei xjlei@snnu.edu.cn 1 School of Computer Science, Shaanxi Normal University, No. 620, West Chang'an Avenue, Xi'an 710119, Shaanxi, China of M dimensions (i.e., objectives).A MOP is called the manyobjective optimization problem (MaOP) when the number of objectives is larger than three (i.e., M > 3).The essential difference between single-objective optimization problems and multi-objective optimization problems is that multi-objective problems cannot have an optimal solution to achieve the best performance of all objectives, but have a set of optimal solutions.Assume x and y are two solutions in the decision space, if x and y satisfy ∀i ∈ 1, . . ., M, f i (x) ≤ f i (y) and ∃i ∈ 1, . . ., M, f i (x) < f i (y), we say x dominates y or y is dominated by x.The Pareto optimal solution is defined as a solution that is not dominated by any other solution.The Pareto set (PS) consists of all Pareto optimal solutions, while the objectives corresponding to the solutions in PS constitute the Pareto front (PF).Evolutionary algorithms and memetic algorithms [7][8][9] play important roles in solving MOPs.Generally speaking, evolutionary algorithms obtain the optimal solution in global search, memetic algorithms improves the quality of the solution through local search.With the development of research, some typical algorithms (NSGA-II [10], MOEA/D [11]) show the superiority of solving MOP and have become a popular multi-objective evolutionary algorithm (MOEA) [12].The aims of MOEAs are to obtain PS that makes each objective as good as possible (i.e., good convergence) and evenly distributed along with the PF (i.e., good diversity).However, when the number of objectives M increases, the performance of MOEA decreases rapidly and the number of Pareto optimal solutions increases exponentially.In this case, the two solutions are easily regarded as non-dominated relations.Moreover, many important scientific problems such as high computational complexity, fewer sampling points and anti-convergence phenomenon (i.e., the solution with poor convergence will make the population diversity good) in high-dimensional objective space need to be studied and solved [13].
At present, the many-objective evolutionary algorithms (MaOEAs) for solving MaOPs can be divided into three categories, but they have different defects.Modified Pareto dominance methods belong to the first category.Traditional Pareto dominance relation loses its effect in the highdimensional objective space due to the strict dominance condition.To relieve this issue, many scholars focus on modifying or relaxing the Pareto dominance rule.For instance, ε-dominance [14] and fuzzy dominance [15] employ modified definitions of dominance to enhance the selection pressure.Gird-dominance [16] by converting function fitness into grid coordinate value, the original non-dominated solutions may become mutually dominant as having the same grid coordinate value in some dimensions, thus expanding the area of the dominant region in the objective space.Zhang et al. [17] employ a knee point-based selection scheme to select non-dominated solutions.Li et al. [18] proposed a shift-based density estimation strategy to make Pareto dominance-based algorithms more suitable for many-objective optimization.Experiments have illustrated that the above algorithms can accelerate the convergence of population, but the improved dominance relationship will lead to the population converging in the sub-region of true PF and losing diversity [19].
Indicator-based MaOEAs form the second category.These MaOEAs use indicators to evaluate the solutions and guide the search process.The hypervolume (HV) [20,21] is one widely used indicator.The HypE employs Monte Carlo simulation to estimate the HV contribution of the candidate solutions and reduce the high complexity of calculating the HV [22].Other popular indicators include the R2 indicator [23] and the inverted generational distance (IGD) [24].The many-objective metaheuristic based on the R2 indicator II (MOMBI-II) algorithm [25] uses the R2 indicator to lead the population during the evolutionary search process.The indicator-based evolutionary algorithm (IBEA) can be integrated with arbitrary indicators [26].Moreover, Other indicators used to extend MOEAs for MaOPs include P [27], pure diversity (PD) [28], and coverage to PF (CPF) [29].However, these approachs have high computational complexity.
The third category is based on decomposition.The main idea is to decompose a MaOP into a set of sub-problems according to the widely distributed weight vectors, and the population is continuously converged to PF by minimizing each sub-problem.On the one hand, some decompositionbased MOEAs such as reference vector guided EA (RVEA) [30] and MOEA based on decomposition (MOEA/D) [11] decompose a MOP into several SOPs.On the other hand, some other decomposition-based MOEAs such as NSGA-III [31] and SPEA [32] based on reference direction decompose a MOP into several simpler MOPs by dividing the objective space into several subspaces.However, these methods have poor effect on irregular PFs.
Although these existing methods have made significant progress in solving MaOPs, there are still some problems.The method of evaluating the diversity of the solution set can directly affect the final performance of the algorithm, However, the characteristics of high-dimension and few sampling points in the high-dimensional objective space make it difficult to judge the diversity of the solution set, and there are also difficulties in quantitatively evaluating the diversity.As we know, the finite solution set can only be very sparsely distributed in high-dimensional space [33], which is called "curse of dimension".Some studies have also pointed out that algorithms always encounter difficulties in the diversity management of many objective optimizations [34,35].Therefore, diversity assessment and management pose severe challenges to many objective optimizations.
To solve the above problems, this article proposes a PC strategy to evaluate the diversity of solution sets to obtain excellent populations.At the same time, a manyobjective optimization evolutionary algorithm based on point crowding-degree is proposed to solve the many-objective optimization problem.Specifically, when evaluating the diversity of the solution set, the PC strategy not only considers the distance between the nearest two points in the objective space, but also considers the difference between each objective function, and considers the influence of the surrounding neighbor points on the diversity of this point.At the same time, the selection of evolutionary operators effectively balances convergence and diversity, and further points to the search area.The main contributions of this paper are summarized as follows: (1) The new point crowding-degree strategy proposed not only considers the distance between any two points as large as possible, but also considers the difference between each dimension component as large as possible, and considers the influence of surrounding neighbor points on the diversity of this point, so as to obtain a better diversity solution set.
(2) The selection of evolutionary operator is designed to balances convergence and diversity, and further points to the search region.Specifically, the minimum solution of the Tchebycheff function value is selected as a parent, and this process also accelerates the convergence.Another parent's random selection in non-dominated solutions also ensures population diversity.(3) A PCEA algorithm is proposed to effectively balance the convergence and diversity of solution sets in manyobjective optimization.
The remainder of this article is organized as follows: Sect. 2 introduces the proposed algorithm in detail; In Sect.3, the experimental results and correlation analysis of the algorithm are given; Finally, the conclusion and future work are given in Sect. 4.

Selection strategy
The selection of evolutionary operators is very important in the whole search process.The selection operator aims to select the parent with good convergence and diversity to generate high-quality offspring.Firstly, the non-dominated solutions are selected by non-dominated sorting in the candidate solutions.Then, some solutions are deleted by the improved Tchebycheff function.Finally, the minimum solution of the Tchebycheff function value is selected as a parent, which also accelerates the convergence.Another parent's random selection in non-dominated solutions also ensures population diversity.We find the T close tweight vectors to each weight vector according to the Euclidean distances of any two weight vectors.Specifically, the function of weight vector λ i can be defined as follows: where λ = (λ 1 , . . ., λ m ) T is a given weight vector, Z = z 1 , . . ., z m is a reference point, The crossover operation of the algorithm uses differential evolution (DE) [36] and simulated binary crossover (SBX) [37] to generate new offspring.The purpose is to further point the search region based on the excellent parent genes.The mutation operator uses polynomial mutation and roulette selection to generate new offspring.Algorithm 1 introduces the details of Cross-mutation operation.

PC strategy
The method to evaluate the diversity of the solution set can directly affect the final performance of the algorithm.Aiming at the characteristics of high dimensional many-objective evolutionary problems with high dimension and fewer sampling points, the PC strategy not only considers the distance between the nearest two points in the objective space but also considers the difference between each objective function.The specific strategies are as follows.Assuming that the set of the current objective space is A = {x 1 , . . ., x n }, , we first calculate the distance between x i and x j at any two points: ) 2 is the sine value of the angle between the straight line x i − x j andL = (L 1 , . . ., L M ) (L k = 1, other components ar e 0).All these sine values are multiplied so that the included angle between x i − x j and L = (L 1 , . . ., L M )(L k = 1, other components ar e 0) is as small as possible (|x i k − x j k | as large as possible), which ensure that each component will be as different as possible, and is a parameter greater than 0.Then, for any point x j , compare the size of the objective value to determine its neighbors by the following: Then calculate the crowding degree of x j : where H represents the size of the set H .The crowding degree C D(x j ) of a point not only considers the Euclidean distance between two points, but also considers the difference between each component, and finally considers the influence the of neighbor solution on its crowding degree.The greater the C D(x j ) value, the better the sparsity of x j .By calculating the crowding degree of each individual, delete the individuals with the smaller crowding value, and then maintain the diversity of the population.Algorithm 2 introduces the details of point crowdingdegree.First, in lines 1-4 of algorithm 2, each solution in the offspring population O is operated in the following two steps: randomly select an individual to determine its neighbor individual according to formula (4); Then calculate the crowing-degree according to formula (5).In lines 6-14 of algorithm2, individuals with larger crowing-degree values will be retained.r represents the selected individual and removes r from O .Perform the while loop until the condition is met.

Steps of the proposed algorithm
Based on the above, a point crowding-degree based evolutionary algorithm for many-objective optimization is proposed, Algorithm 3 introduces the overall framework of PCEA.First, the initialization operation is performed.In the main loop of the evolutionary search process, the non-dominated sorting is performed in the candidate solution, and the Tchebycheff function value of each individual is calculated according to Eq. ( 2) and stored in the set F. Selects the neighbors of the solution based on T neighboring weight vectors.The crossover and mutation operation (i.e., Algorithm 1) is performed to generate a new offspring population O .Then, update Z : for Subsequently, the external population is updated according to the point crowdingdegree strategy (i.e., algorithm 2).By calculating the crowding-degree of each individual, the individuals with small crowding degree are deleted, and then the diversity of the population is maintained.Repeat the evolutionary search process until the stop criteria are met, and finally output the final population O. 3 Computational studies and results

Experimental setting
In this section, the proposed algorithm is compared with four state-of-the-art algorithms on 15 multi-objective benchmark functions (MaF) in the CEC2018 MaOP competition, such as NSGA-III [31], MOEA/DD [38], KnEA [17], RVEA [30], NSGA-II-SDR [39], MSEA [40], GrEA [16] and VaEA [41].The following comparison experiment uses the DTLZ (DTLZ1-DTLZ7) test function.Each problem will be tested for 5, 10 and 15 objectives.NSGA-III [31] supplies and updates well-spread reference points adaptively to maintain the diversity among population members.MOEA/DD [38] combines dominance and decomposition-based approaches to balance the convergence and diversity of the evolutionary process.KnEA [17] is a knee point-driven EA to solve MaOPs.RVEA [30] adopts a scalarization approach named angle-penalized distance to balance convergence and diversity.NSGA-II-SDR [39] proposes a new dominance relation to better balance convergence and diversity for evolutionary many-objective optimization.MSEA [40] proposes a multistage MOEA for better diversity performance to address the limitations of existing diversity preservation approaches.
GrEA [16] is to exploit the potential of the grid-based approach to strengthen the selection pressure toward the optimal direction while maintaining an extensive and uniform distribution among solutions.VaEA [41] suggests a vector angle-based evolutionary algorithm for unconstrained (with box constraints only) many-objective optimization problems, and the maximum-vector-angle-first principle is used in the environmental selection to guarantee the wideness and uniformity of the solution set.
For these four compared algorithms, all data of each algorithm given in this section are averaged over 30 independent runs for each test case on PlatEMO [42].In PCEA, Differential evolution (DE) and simulated binary crossover (SBX) are used; distribution index is 20 and crossover probability is 1 in the SBX operator; CR is 0.5 and F is 0.5 in the DE operator; In this experiment, θ in PC strategy is 0.5; the size of neighborhood list T is set to 0.1N; J is set to 0.9.The settings of the experimental studies in this article are identical to the standard for the CEC2018 MaOP competition, which can be found in [43], together with the details of the benchmark functions.The algorithms are implemented by using the MATLAB language on a PC with Intel(R) Core (TM) i5-7500 CPU @ 3.40GHz 3.41 GHz (Windows 10 operating system).

Performance metrics
In this article, the inverted generational distance (IGD) [44] is used to quantitative measurement the performances of algorithms.IGD can be calculated by: where S is the solution set obtained by the algorithm, S * is composed of evenly distributed reference points sampled from true PF, and dist(x, y) denotes the Euclidean distance between solution yin S and solution x in S * *.IGD measures the average minimum distance from each solution from S * to S. For an algorithm, a smaller IGD value means a better quality of the objective vectors of obtained solutions for approximating the PF.The benefits of IGD lie in its computational efficiency and generality for measuring both convergence and diversity of solutions.IGD requires a set of reference Pareto optimal solutions.Roughly 10,000 points uniformly sampled on the Pareto fronts are used in the calculation of IGD for each test problem.Wilcoxon Rank-Sum test [45] is used in the sense of statistics to compare the mean IGD of the compared algorithms.It tests whether the performance of PCEA on each test problem is better ("+"), same ("="), or worse ("−") than that of the compared algorithms at a significance level of 0.05 by a two-tailed test.

Experimental results and comparison
In this article, the proposed PCEA algorithm is compared with other 8 algorithms on 45 MaF test problems and 12 DTLZ test problems.For each test problem, the result with the best performance is marked in bold.F1-k represents that the number of objectives adopted in F1 is k."+" means that PCEA outperforms its competitor algorithm, "−" means that PCEA is worse than its competitor algorithm, and "=" means that the competitor algorithm has the same performance as PCEA.For the 8 problems (F1, F2, F4, F5, F7, F8, F9 and F15) with partial PFs whose projections do not fully cover the unit hyperplane, the mean values of IGD obtained by PCEA are smaller than those obtained by NSGA-III, MOEA/DD, KnEA, RVEA, NSGA-II-SDR and MSEA on 20, 20, 15, 20, 15 and 20 problems, which indicate that the proposed algorithm obtains the best overall performance in the form of IGD on most problems.For the problem F6 with degraded PF, the performances of PCEA outperform NSGA-III, MOEA/DD, KnEA, RVEA, NSGA-II-SDR and MSEA on 3 problems.These comparison results demonstrate that the PC strategy brought versatility for the diverse PFs.

MaF test suite
When dealing with problems with PF projection fully covering the unit hyperplane (F3, F10, F11, F12, F13 and F14), the mean values of IGD obtained by PCEA are smaller than those obtained by NSGA-III, MOEA/DD, KnEA, RVEA, NSGA-II-SDR and MSEA on10, 8, 8, 8, 9 and 9 problems, which imply that PCEA obtains the best overall performance in the form IGD on most problems.
In order to more intuitively observe the ability of the seven algorithms to balance convergence and diversity on the MaF test suite.Figures 1 and 2 give the solution sets obtained by seven algorithms of 3-objective MaF1 and 10-objective MaF8 respectively.Because the PF of 10-objective MaF8 cannot be displayed through space, parallel coordinates are used.For 3-objective MaF1, PF is a three-dimensional inverted triangle.As shown in Fig. 1, although the solution sets obtained by the seven algorithms can successfully converge to the corresponding objective dimensions on PF, their diversity is significantly different.The convergence and diversity of the solution set obtained by MSEA are similar to those of PCEA, but the middle part is not as good as PCEA.The distribution uniformity of NSGAIII, MOEAD, KnEA, RVEA and NSGA-II-SDR is slightly worse, and the overall performance is lower than that of PCEA.In general, the solution set obtained by PCEA is more uniform and compact, and is superior to the other six MaOEAs in terms of convergence and diversity.
As shown in Fig. 2, the solution set obtained by PCEA is superior to the six comparative MaOEAs in terms of convergence and diversity.MOEA/DD and RVEA cannot solve this problem, and the solution set does not converge.Compared with NSGAIII, KnEA, NSGA-II-SDR and MSEA, the lines of the solution set obtained by PCEA are more compact and uniform, and it can best characterize the PF of 10-objective MaF8.Therefore, PCEA performs best among these MaOEAs.4, the statistical performance of PCEA on the 5 problems is better than that of the comparison algorithm.Among them, there is 1 problem for 5-objective, and there are 2 problems for 10-objective and 15-objective, indicating that PCEA is better at dealing

Ablation experiment
To verify the validity of the second term of the proposed distance method (i.e., Eq. 3) the ablation study is carried out.The verification process only uses Eq. ( 3) for simple diversity calculation, and does not use other environmental selection strategies to avoid the impact on the final population.D1 only uses the first term of the Eq.(3) (i.e., remove the second term), and D2 uses all of Eq. (3).In Table 5, the PD mean and standard deviation values obtained by D1 and D2 in 30 independent runs on 5,10,15 objective test benchmarks are reported.For each test problem, the results with the best performance are marked in bold.Pure Distance (PD) proposed by Wang et al. should measure diversity by calculating the dissimilarity between solution x and solution set S for high-dimensional objective optimization problems, which is defined as follows:  where the calculation process of dissimilarity can be referenced.The larger the PD, the better the diversity of the solution set.
It can be seen from Table 5 that D2 is significantly better than D1 in 5, 7 and 5 test questions respectively.It can be seen that the second term is essential to improve the diversity of the population, because the second term considers the differences between each component.

Extending to other MaOEAs
As mentioned above, the PC strategy is a good strategy to evaluate population diversity.This section verifies its effectiveness by extending the PC strategy to other MaOEAs.To this end, it is integrated into NSGA-III and KnEA to update the external population, and the resulting variants are called NSGA-III-PC and KnEA-PC, respectively.Table 6 reports the calculation results of the algorithm on 5, 10, 15 objective test benchmarks.The IGD mean and standard deviation values of 41 MOEAs obtained in 30 independent runs are reported in the table.The parameter setting is the same as that used in the corresponding method.For each test problem, the results with the best performance are marked in bold.F1-k indicates that the number of objectives used in the F1 test problem is k.As shown in Table 6, in 21 instances of the DTLZ test function, NSGA-III-PC obtained significantly better results than NSGA-III by comparing IGD values.NSGA-III-PC was significantly better than NSGA-III in 16 instances.Similarly, KnEA-PC also achieved significantly better results than KnEA, with KnEA significantly better than KnEA in 13 instances.These results and analysis show that the PC strategy is a general and effective method, which is beneficial to improve the performance of other MaOEAs algorithms.

Analysis of PC strategy
The core of this article is the PC strategy, which is used to evaluate the diversity of solution sets in high-dimensional objective space.In Sect.3.4, the ablation study proves the effectiveness of the distance formula (Eq.3).In Sect.3.5, the PC strategy is extended to other MaOEAs.Experiments show that the extended PC strategy algorithm is better than the original algorithm.This is also consistent with the advantages of the PC strategy itself, because it not only considers the distance between the nearest two points in the objective space, but also considers the difference between each objective function when evaluating the diversity of the solution set.Compared with most other algorithms, the PC strategy does not require a grid environment when evaluating the diver-   +/=/-15/0/6 -13/2/6 -Bold marks indicate the best performing results.F1-k represents that the number of objectives adopted in test problem F1 is k sity of solution sets and is independent of any parameters or references.Using parameters or references may reduce objectivity.Therefore, the PC strategy is a simple and general strategy that can be integrated into other algorithms to improve the diversity of algorithms in high-dimensional objective space.

Conclusion
In this article, aiming at the existing problems of manyobjective optimization in the high dimensional objective space, a point crowding-degree based evolutionary algorithm for many-objective optimization is proposed to obtain a set of solutions with good diversity and convergence.Specifically, when evaluating the diversity of solution set, the point crowding degree strategy not only considers the distance between the nearest two points in the objective space but also considers the difference between each objective function.At the same time, the selection of evolutionary operators balances convergence and diversity, and further points to the search region.The PCEA is compared experimentally with several state-of-the-art algorithms and the experimental results show that the proposed PCEA algorithm has strong competitiveness and better overall performance.In addition, the proposed PC strategy is integrated into other advanced MaOPs methods.The results show that it is beneficial to improve the performance of other MaOEAs algorithms.
In the future, the point crowding strategy can be combined with other advanced algorithms to solve many-objective optimization problems in the high dimensional objective space.

2 :
Require: O (offspring population) Ensure: O(final population) 1: While O is not full do For each solution in O do 3: Select an individual to determine its neighbors by formula (4) 4: Calculate the crowding degree value of individual by formula (5) 5: End 6: Find the individual r =arg max(C D(x)) to O 7: Remove r from O 8: End while 9: return O

Table 1
Comparison of the IGD value (mean and standard deviation) of the seven algorithms on the MaF test problems

Table 4
reports the IGD mean and standard deviation values obtained by 7 MOEAs on 12 DTLZ test problems.On the 12 test problems in Table

Table 2
Comparison of the IGD value (mean and standard deviation) of the seven algorithms on the MaF test problems

Table 3
Comparison of the IGD value (mean and standard deviation) of the seven algorithms on the MaF test problems

Table 4
Comparison of the IGD value (mean and standard deviation) of the seven algorithms on the DTLZ test problems

Table 5
Comparison of the PD value (mean and standard Deviation) of D1 and D2 on the DTLZ test problems "+" means that D2 outperforms D1, "−" means that is worse than D1, and "=" means that the D2 has the same performance as D1

Table 6
Comparison