Nonelitist Evolutionary Multiobjective Optimizers Revisited
Abstract.
Since around 2000, it has been considered that elitist evolutionary multiobjective optimization algorithms (EMOAs) always outperform nonelitist EMOAs. This paper revisits the performance of nonelitist EMOAs for biobjective continuous optimization when using an unbounded external archive. This paper examines the performance of EMOAs with two elitist and one nonelitist environmental selections. The performance of EMOAs is evaluated on the biobjective BBOB problem suite provided by the COCO platform. In contrast to conventional wisdom, results show that nonelitist EMOAs with particular crossover methods perform significantly well on the biobjective BBOB problems with many decision variables when using the unbounded external archive. This paper also analyzes the properties of the nonelitist selection.
1. Introduction
Since no solution can simultaneously minimize multiple conflicting objective functions in general, the ultimate goal of multiobjective optimization problems (MOPs) is to find a Pareto optimal solution preferred by a decision maker (Miettinen, 1998). When the decision maker’s preference information is unavailable a priori, an “a posteriori” decision making is performed. The decision maker selects the final solution from a solution set that approximates the Pareto front in the objective space.
An evolutionary multiobjective optimization algorithm (EMOA) is frequently used to find an approximation of the Pareto front for the “a posteriori” decision making (Deb, 2001). A number of EMOAs have been proposed in the literature. Classical EMOAs include VEGA (Schaffer, 1985), MOGA (Fonseca and Fleming, 1993), and NSGA (Srinivas and Deb, 1994) proposed in the 1980s and 1990s. They are nonelitist EMOAs, which do not have a mechanism to maintain nondominated solutions in the population. Some elitist EMOAs (e.g., SPEA (Zitzler and Thiele, 1999), SPEA2 (Zitzler et al., 2001), and NSGAII (Deb et al., 2002a)) have been proposed in the early 2000s. Elitist EMOAs explicitly keep nondominated solutions found during the search process.
Some EMOAs store nondominated solutions found so far in an unbounded or bounded external archive independently from the population. For example, MOGLS (Ishibuchi and Murata, 1998) proposed in the mid1990s does not maintain elite solutions in the population but stores all nondominated solutions found so far in the unbounded external archive. MOEA (Deb et al., 2005) stores nondominated solutions in the population and nondominated solutions in the unbounded external archive. PESA (Corne et al., 2000) uses the nonelitist population and the elitist bounded external archive. The external archive in these EMOAs (e.g., MOGLS, MOEA, and PESA) plays two roles. The first role is to provide nondominated solutions found so far to the decision maker. The performance of these types of EMOAs is also evaluated based on solutions in the external archive, rather than the population. The second role is to perform an elitist search. For example, parents for mating are selected from the external archive in PESA. Some elitist individuals in the external archive can enter the population in MOGLS. Since these types of EMOAs explicitly exploit elitist solutions as explained above, they can be categorized into elitist EMOAs.
Apart from algorithm development, the external archive has been used only for the first role (e.g., (Fonseca and Fleming, 1993; LópezIbáñez et al., 2011; Bringmann et al., 2014; Brockhoff et al., 2015; Wessing et al., 2017)). As pointed out in (Bringmann et al., 2014), good potential solutions found so far are likely to be discarded from the population. The external archive that stores all nondominated solutions independently from EMOAs can address this issue. The external archive for the first role can be incorporated into all EMOAs without any changes in their algorithmic behavior. The external archive is useful for realworld problems where the evaluation of each solution is expensive, i.e., the total number of examined solutions is limited, and the archive maintenance cost is relatively small in comparison with the solution evaluation cost. If the decision maker wants to examine a small number of nondominated solutions, solution selection methods are available such as hypervolume indicatorbased selection methods (e.g., (Bringmann et al., 2014)) and distancebased selection methods (e.g., (Singh et al., 2019)).
This paper revisits nonelitist EMOAs with the unbounded external archive only for the first role (performance evaluation). When the performance of EMOAs is evaluated based on solutions in the external archive as in (LópezIbáñez et al., 2011; Bringmann et al., 2014; Brockhoff et al., 2015; Wessing et al., 2017), the role of EMOAs is only to find nondominated solutions with high quality. Thus, EMOAs do not need to maintain nondominated solutions found so far in the current population with the population size . We investigate three environmental selections: bestall (BA), bestfamily (BF), and bestchildren (BC). While BA and BF are elitist selections, BC is a nonelitist selection. Although BA is a traditional selection, BF and BC restrict a selection only among parents and children. Thus, nonparents do not directly participate in the selection process in BF and BC unlike traditional  and selections. In BC, all parents are removed from the population regardless of their quality. Then, the topranked out of children enter the population. Subsection 2.3 explains BA, BF, and BC in detail. We examine the performance of EMOAs with the three selections on the biobjective BBOB problem suite (Tusar et al., 2016). We use five crossover methods and four ranking methods in representative EMOAs.
Our contributions in this paper are at least threefold:

We demonstrate that the nonelitist BC selection performs significantly well on the biobjective BBOB problems with many decision variables when using the unbounded external archive. Although most EMOAs proposed in the 2000s are elitist EMOAs, our results indicate that efficient nonelitist EMOAs could be designed. Thus, our results significantly expand the design possibility of EMOAs.

We demonstrate that restricted replacements in BF and BC are suitable for crossover methods with the preservation of statistics (Kita et al., 1998) (e.g., the property where the covariance matrix of children is the same as that of the parents) such as SPX (Tsutsui et al., 1999) and REX (Akimoto et al., 2009).

We discuss why the simple BA selection performs worse than the restricted BF and BC selections. We also analyze the properties of the nonelitist BC selection.
The rest of this paper is organized as follows. Section 2 provides some preliminaries of this paper, including the definition of MOPs, the five crossover methods, and the three environmental selections. Section 3 describes the experimental setup. Section 4 examines the performance of the three environmental selections. Section 5 concludes this paper with discussions on future research directions.
2. Preliminaries
2.1. Definition of continuous MOPs
A continuous MOP is to find a solution that minimizes a given objective function vector . Here, is the dimensional solution space, and is the dimensional objective space. is the number of decision variables, and is the number of objective functions.
A solution is said to dominate iff for all and for at least one index . If is not dominated by any other solutions in , is a Pareto optimal solution. The set of all is the Pareto optimal solution set, and the set of all is the Pareto front. The goal of MOPs for the “a posteriori” decision making is to find a nondominated solution set that approximates the Pareto front in the objective space.
2.2. Crossover methods in realcoded GAs
We use the following five crossover methods in realcoded GAs: simulated binary crossover (SBX) (Deb and Agrawal, 1995), blend crossover (BLX) (Eshelman and Schaffer, 1992), parentcentric crossover (PCX) (Deb et al., 2002b), simplex crossover (SPX) (Tsutsui et al., 1999), and realcoded ensemble crossover (REX) (Akimoto et al., 2009). Here, we briefly explain the five crossover methods.
Traditional GAs use two variation operators: crossover and mutation. In contrast, realcoded GAs with BLX, PCX, SPX, and REX do not need the mutation operator because they can generate diverse children by adjusting their control parameters (e.g., the expansion rate in SPX). However, the polynomial mutation (PM) (Deb and Agrawal, 1995) is applied to two children generated by SBX in most studies. In other words, SBX and PM have been considered to be a set. For this reason, we apply PM to children generated only by SBX. We refer to “SBX and PM” as “SBX” for simplicity.
Table 1 shows the properties of the five crossover methods. Although SBX and BLX are traditional twoparent crossover methods, PCX, SPX, and REX are multiparent crossover methods. PCX, SPX, and REX are rotationally invariant. The performance of EMOAs with rotationally invariant operators does not depend on the coordinate system. While PCX and REX use a Normal probability distribution, BLX and SPX use a uniform probability distribution. The probability distribution used in SBX is unclear. Although the center of the distribution of children is the mean vector of parents in BLX, SPX, and REX, that is one of the parents in SBX and PCX. SPX and REX have a property called the “preservation of statistics” proposed in (Kita et al., 1998). In a crossover method with this property, children inherit the statistics (e.g., the mean vector and the covariance matrix) from their parents.
Cent.  Prob.  Rot.  Sta.  Parameters  

SBX  parent  ?  ,  
BLX  mean  U  
PCX  parent  N  ,  
SPX  mean  U  
REX  mean  N 
Figure 1 shows the distribution of children generated by the five crossover methods. SBX simulates the working principle of the singlepoint crossover in binarycoded GAs. Since SBX is a variablewise operator, most children are generated along the coordinate axes. The distribution of children is controlled by in SBX (and in PM). In BLX, the th element () of a child is uniformly randomly selected from the range . Here, and . and are parents, and is the expansion factor.
PCX is a parentcentric version of UNDX (Kita et al., 1999), which is a multiparent extension of unimodal normal distribution crossover (UNDX) (Ono and Kobayashi, 1997). While the center of the distribution of children is the mean vector of parents in UNDX, that is one of the parents in PCX. PCX requires two parameters and that control the variances of two Normal distributions. SPX can be viewed as being a rotationally invariant version of BLX. SPX uniformly generates children inside an expanded simplex formed by parents. The theoretical analysis presented in (Higuchi et al., 2000) shows that SPX with the expansion factor satisfies the preservation of statistics. REX is a generalized version of UNDX. REX using a zeromean Normal distribution with the variance satisfies the preservation of statistics (Akimoto, 2010).
2.3. Environmental selections
We consider a “simple” EMOA shown in Algorithm 1.
After the initialization of the population with the population size (line 1), the following operations are repeatedly performed until a termination condition is satisfied.
First, parents are randomly selected from such that their indices are different from each other (line 3).
Let be a set of the parents.
Then, children are generated by applying a crossover method to the same parents times (line 4).
Below, we explain the following three environmental selections: bestall (BA), bestfamily (BF), and bestchildren (BC). Note that our main contributions in this paper are analysis of BA, BF, and BC in Section 4, not proposing BA, BF, and BC. Algorithms 2, 3, and 4 show BA, BF, and BC, respectively. While BA and BF are elitist selections, BC is a nonelitist selection. The three selections require a method of ranking individuals based on their quality. Similar to MOCMAES (Igel et al., 2007), BA, BF, and BC can be combined with any ranking method. In this paper, we use four ranking methods in NSGAII (Deb et al., 2002a), SMSEMOA (Beume et al., 2007), SPEA2 (Zitzler et al., 2001), and IBEA with the additive indicator (Zitzler and Künzli, 2004). We denote their ranking methods as “NS”, “SM”, “SP”, and “IB”, respectively. Individuals are ranked based on their nondomination levels in NS and SM. The tiebreakers are the crowding distance in NS and the hypervolume contribution in SM. In SP and IB, individuals are sorted based on their socalled fitness values in descending order. In this paper, XYZ represents the EMOA (Algorithm 1) with an environmental selection X, a crossover method Y, and a ranking method Z. For example, BASBXNS is the EMOA with BA, SBX, and NS.
In BA (Algorithm 2), the topranked individuals are selected from the union of and . BA is the traditional elitist selection used in most EMOAs (e.g., NSGAII and SPEA2). It should be noted that BASBXNS is not identical to NSGAII. The differences between BASBXNS and NSGAII are the random parent selection and the children generation. The same parents are used to generate children in BA. For the same reason, BASBXSP, BASBXSM, and BASBXIB are not identical to SPEA2, SMSEMOA, and IBEA, respectively.
In BF (Algorithm 3), the environmental selection is performed only among the socalled “family” that consists of children in and parents in . After all individuals in the union of and have been ranked, only parents in are removed from . Then, the best individuals are selected from the union of and . Although nonparents in do not directly participate in the selection process, they contribute to assign ranks to individuals in the union of and . While the maximum number of individuals replaced by children is in BA, that is in BF. Since only parents can be replaced by children in BF, nonparents can survive to the next iteration with no comparison. Selections among families as in BF are used in GAs for singleobjective optimization (e.g., the deterministic crowding (Mahfoud, 1992)).
In BC (Algorithm 4), the environmental selection is performed among children in . We assume that . After parents in have been removed from , all individuals in the union of and are ranked. Then, the best individuals are selected from . Since all parents are deleted from regardless of their quality, BC does not maintain nondominated individuals in . Thus, BC is a nonelitist selection in contrast to the elitist BA and BF selections. While individuals in are replaced with children in in most classical EMOAs (e.g., MOGA), only parents in are replaced with the best out of children in in BC. Thus, BC is different from the traditional selection.
BC can be viewed as being an extension of just generation gap (JGG) (Akimoto, 2010) to multiobjective optimization. JGG is an environmental selection in GAs for singleobjective continuous optimization. The only difference between BC and JGG is how to assign ranks to individuals. Individuals are ranked based on their objective values in JGG and their objective vectors in BC. The results presented in (Akimoto, 2010) show that nonelitist GAs with JGG significantly outperform elitist GAs on singleobjective test problems (especially multimodal problems) when using crossover methods with the preservation of statistics.
3. Experimental settings
We conducted all experiments using the comparing continuous optimizers (COCO) platform (Hansen et al., 2016). COCO is the standard platform used in the black box optimization benchmarking (BBOB) workshops held at GECCO (2009–present). We used the latest COCO software (version 2.2.2) downloaded from https://github.com/numbbo/coco. COCO provides six types of BBOB problem suites, including the singleobjective BBOB noiseless problem suite (Hansen et al., 2009). The biobjective BBOB problem suite (Tusar et al., 2016) consists of 55 biobjective test problems designed based on the idea presented in (Brockhoff et al., 2015). Each biobjective BBOB problem is constructed by combining two singleobjective BBOB problems. For example, the first and second objective functions of are the Sphere function and the rotated Rastrigin function, respectively. The number of decision variables is . For details of the 55 biobjective test problems, see (Tusar et al., 2016). For each problem, runs were performed. These settings adhere to the analysis procedure adopted by the GECCO BBOB community. The maximum number of function evaluations was set to .
COCO also provides the postprocessing tool that aggregates experimental data. COCO automatically stores all nondominated solutions found by an optimizer in the unbounded external archive. The performance indicator (Brockhoff et al., 2016) in COCO is mainly based on the hypervolume value of nondominated solutions in the unbounded external archive. When no solution in the external archive dominates a predefined reference point in the normalized objective space, the value is calculated based on the distance to the socalled region of interest. For details of , see (Brockhoff et al., 2016).
We implemented all algorithms using jMetal (Durillo and Nebro, 2011). Source codes of all algorithms are available at https://sites.google.com/view/nemorgecco2019/. For all five crossover methods (except for PCX), we used the control parameters recommended in the literature shown in Table 1. Since PCX with performed poorly in our preliminary study, we set to similar to SPX and REX. For comparison, we evaluated the performance of the original NSGAII, SPEA2, SMSEMOA, and IBEA. SBX and PM with , , , and were used in the original EMOAs. As in (Tusar and Filipic, 2016), was set to . The number of children was set to . We set the value based on our preliminary results and studies of GAs for singleobjective optimization (e.g., (Akimoto et al., 2009; Akimoto, 2010)).
4. Results
This section shows analysis of the three environmental selections (BA, BF, and BC). Since SPX is suitable for BF and BC, we mainly discuss results of EMOAs with SPX. Although results of EMOAs with REX are similar to those with SPX, we do not show them here due to space constraint. As shown in Subsection 4.4, SBX, BLX, and PCX are not suitable for BA, BF, and BC.
Subsection 4.1 shows a comparison among BASPXNS, BFSPXNS, BCSPXNS, and the original NSGAII. Subsection 4.2 investigates why BA performs poorly. Subsection 4.3 analyzes the advantages and disadvantages of the nonelitist BC compared with the elitist BF. Subsection 4.4 examines the performance of BA, BF, and BC with other crossover methods (SBX, BLX, PCX, and REX). Subsection 4.5 presents a comparison of BA, BF, and BC with other ranking methods (SP, SM, and IB).
4.1. Comparison of BA, BF, and BC
Figure 2 shows results of the original NSGAII, BASPXNS, BFSPXNS, and BCSPXNS on all 55 BBOB problems with . Due to space constraint, results for are not shown, but they are similar to results for . In this section, we use the SPX crossover and the NS ranking method. In Figure 2, “best 2016” is a virtual algorithm portfolio that is constructed from the performance data of 15 algorithms participating in the GECCO BBOB 2016 workshop. Note that “best 2016” does not mean the best optimizer among the 15 algorithms.
Figure 2 shows the bootstrapped empirical cumulative distribution (ECDF) of the number of function evaluations (FEvals) divided by (FEvals/) for 58 target indicator values for all 55 BBOB problems with each . We used the COCO software to generate all ECDF figures in this paper. In Figure 2, the vertical axis indicates the proportion of target indicator values which a given optimizer can reach within specified function evaluations. For example, in Figure 2 (b), BFSPXNS reaches about 40 percent of all 58 target indicator values within evaluations on all 55 problems with in all runs. If an optimizer finds all Pareto optimal solutions on all 55 problems in all runs, the vertical value becomes 1. More detailed explanations of the ECDF (including illustrative examples) are found in (Brockhoff et al., 2015, 2016).
Statistical significance is also tested with the ranksum test () for a given target value using the COCO software. However, statistical test results are almost consistent with ECDF figures. Additionally, the space of this paper is limited. For these reasons, we show only ECDF figures. The statistical test results and other ECDF figures are available at https://sites.google.com/view/nemorgecco2019/.
Figure 2 shows that BASPXNS performs the best until evaluations for . However, the increase of deteriorates the performance of BASPXNS. The evolution of BASPXNS clearly stagnates for . The original NSGAII is the best performer in the early stage for . BFSPXNS and BCSPXNS perform better than NSGAII and BASPXNS in the later stage for all . Interestingly, the nonelitist BCSPXNS performs the best in the later stage for . Although it has been believed that elitist EMOAs always outperform nonelitist EMOAs for about two decades, our results show that the nonelitist BCSPXNS performs better than the elitist NSGAII, BASPXNS, and BFSPXNS on the biobjective BBOB problems with when using the unbounded external archive.
Note that BCSPXNS is not always the best optimizer on all 55 BBOB problems with . Figure 3 shows results on and with . While BFSPXNS outperforms BCSPXNS on , BCSPXNS outperforms BFSPXNS on . Similar to Figure 3, the best optimizer is different depending on the test problem. We attempted to clarify which problem groups BC performs the best (e.g., BC has the best performance on multimodal problems with weak global structure such as and ). Unfortunately, we could not find such a result. An indepth analysis is needed to understand on which problems BC performs well or poorly.
4.2. Why does BA perform poorly?
Here, we discuss the poor performance of BASPXNS observed in Subsection 4.1. The biased distribution of children is likely to cause the poor performance of BASPXNS. As shown in Figure 1 (d), SPX generates children inside a simplex formed by parents. If the parents are close to each other in the solution space, their children are likely to be in local area. If nonparents in the population are ranked worse than the children, the nonparents are replaced with the children in BA. This means that nonparents in notwellexplored area cannot survive to the next iteration. Thus, BASPXNS is likely to lose diversity in the solution and objective spaces as the search progresses.
One may think that the abovementioned issue caused by the biased distribution of children can be addressed by setting to a small value. Figure 4 shows BASPXNS with on all 55 BBOB problems with . In Figure 4, “” is identical to BASPXNS in Figure 2. Figure 4 also shows the results of NSGAII, BFSPXNS and BCSPXNS derived from Figure 2. Figure 4 shows that the performance of BASPXNS can be improved by setting to a small value. However, BASPXNS with any is outperformed by NSGAII, BFSPXNS, and BCSPXNS at the later stage.
In general, a large enough number of children are necessary to find better solutions in the current search area (Akimoto, 2010). Thus, BA is in a dilemma. A large value is helpful for BA to exploit the current search area, but it causes premature convergence. A small value can prevent BA from the premature convergence, but it is not sufficiently large to exploit the current search area. In addition to SPX, we observed the same issue in other crossover methods (except for SBX).
In contrast to BA, only parents can be replaced with children in BF and BC. This restricted replacement in BF and BC can help the population to maintain the diversity. Even if nonparents in notwellexplored area are dominated by the children, the nonparents can survive to the next iteration with no comparison. Thus, BF and BC can address the BA’s dilemma. In fact, BFSPXNS and BCSPXNS perform significantly better than BASPXNS.
4.3. Advantages and disadvantages of BC
As shown in Subsection 4.1, the nonelitist BC performs better than the elitist BF for . Here, we discuss the advantages and disadvantages of BC compared with BF.
Figure 5 (a) shows raw indicator values of the population in BFSPXNS and BCSPXNS on with , which consists of two rotated Rastrigin function instances.
In all 55 BBOB test problems, can be viewed as being a representative multimodal problem.
We slightly modified the COCO software to calculate the value of the population (not the external archive).
A lower raw value is better.
The range of the value in Figure 5 (a) is limited to in order to focus on the interesting behavior of BCSPXNS.
Although the value of the elitist BFSPXNS almost
Figure 5 (b) shows the cumulative number of parents replaced by children. In BFSPXNS, the evolution of clearly stagnates after function evaluations. This result means that BFSPXNS rarely generates better children than parents. In fact, the raw value of BFSPXNS is not significantly improved after function evaluations, as shown in Figure 5 (a). Since BCSPXNS always replaces parents with the best out of children for every iteration, linearly increases. Thus, the replacement of individuals in BC occurs more frequently than that in BF. This property of BC is helpful for exploration of the search space.
The above observations indicate that BC has a similar advantage to simulated annealing (Kirkpatrick et al., 1983), which can move to a worse search point. As pointed out by Deb and Goel (Deb and Goel, 2001), if an elitist EMOA prematurely converges to local Pareto optimal solutions, it is very likely to stagnate. Unless the elitist EMOA finds better solutions far from the current search area, it cannot escape from local Pareto optimal solutions. In contrast, the nonelitist BC always replaces parents with children regardless of the quality of parents. While most elitist environmental selections accept only “downhill” moves on minimization problems, the nonelitist BC can accept “uphill” moves as in simulated annealing. The uphill moves in BC help the population to escape from local Pareto optimal solutions on some multimodal problems.
However, BC has at least two disadvantages compared with the elitist BF. First, as discussed in Subsection 4.1, BC performs worse than BF on some problems even with . Second, as reported in Subsection 4.1, BC performs worse than BF at the early stage. Since BC can accept “uphill” moves as in simulated annealing, the exploitative ability of BC is worse than that of BF. A deterministic or adaptive method of switching BC and BF may be promising to exploit their advantages.
4.4. Which crossover methods are suitable for the nonelitist BC?
The results in Subsection 4.1 show that BCSPXNS outperforms BASPXNS, BFSPXNS, and NSGAII for . Here, we examine which crossover methods are suitable for BC. We are not interested in which crossover method is best. Even though BCSPXNS outperforms BCPCXNS, it does not mean that SPX performs better than PCX. It only means that SPX is more suitable for BC than PCX.
Figure 6 shows results of the three selections with SBX, BLX, PCX, and REX on all 55 BBOB problems with . Due to space constraint, only results for are shown here. The NS ranking method is used in BA, BC, and BF. Figure 6 (a) shows that BASBXNS outperforms BFSBXNS and BCSBXNS. This good performance of BASBXNS is inconsistent with the results in Subsection 4.1. Since SBX can generate children far from their parents as shown in Figure 1, the distribution of children discussed in Subsection 4.2 does not significantly influence the performance of BA. However, BASBXNS performs worse than NSGAII. Figure 6 (b) and (c) show similar results. The evolution of the three selections with BLX and PCX clearly stagnates. Figure 6 (d) shows that results with REX are consistent with the results with SPX. BCREXNS is the best optimizer at the later stage. BFREXNS also performs better than NSGAII.
In summary, SPX and REX are suitable for BC and BF, while SBX, BLX, and PCX are not suitable for them. These results indicate that crossover methods with the preservation of statistics are suitable for BC (and BF). As shown in Table 1, only SPX and REX satisfy the preservation of statistics among the five crossover methods. The results presented in (Akimoto, 2010) show that SPX and UNDX (a special version of REX) are suitable for JGG (a similar selection to BC) in GAs for singleobjective continuous optimization. Interestingly, our results on continuous MOPs are consistent with the results on singleobjective continuous optimization problems. A similarity analysis between singleobjective optimizers and multiobjective optimizers as in (Wessing et al., 2017) may be interesting.
4.5. Comparison of BA, BF, and BC with other ranking methods
We used the NS ranking method in Subsection 4.1. We investigate whether similar results can be obtained when using the SP, SM, and IB ranking methods (see Subsection 2.3).
Figure 7 shows the comparison of BA, BF, and BC with SP, SM, and IB for . We do not show results for , but they are similar to the results in Subsection 4.1. SPX is used as a crossover method. Figures 7 (a), (b), and (c) also show results of the original SPEA2, SMSEMOA, and IBEA, respectively.
Figure 7 shows that results with SP, SM, and IB are consistent with the results with NS. BF and BC outperform the original SPEA2, SMSEMOA, and IBEA at the later stage. BC is the best optimizer at the later stage. The poor performance of BA can be observed in Figure 7. Our results show that the relative performance of BA, BF, and BC does not significantly depend on the choice of a ranking method.
5. Conclusion
We examined the effectiveness of the two elitist selections (BA and BF) and the nonelitist selection (BC) on the biobjective BBOB problem suite. We used five crossover methods and four ranking methods. For about two decades, it has been considered that elitist EMOAs always outperform nonelitist EMOAs. Interestingly, our results show that the nonelitist BC performs better than the two elitist selections and the four original EMOAs (NSGAII, SPEA2, SMSEMOA, and IBEA) on the biobjective BBOB problems with many decision variables when using the unbounded external archive and a crossover method with the preservation of statistics (i.e., SPX and REX). The choice of a ranking method does not significantly influence the relative performance of BC. We also analyzed the advantages and disadvantages of the nonelitist BC selection.
A number of interesting directions for future work remain. Although only elitist EMOAs have been studied in the 2000s, our results indicate that efficient nonelitist EMOAs could be realized. Designing nonelitist versions of MOES (Wessing et al., 2017) and MOCMAES (Igel et al., 2007) based on BC may be promising.
Acknowledgments
This work was supported by National Natural Science Foundation of China (Grant No. 61876075), the Program for Guangdong Introducing Innovative and Enterpreneurial Teams (Grant No. 2017ZT07X386), Shenzhen Peacock Plan (Grant No. KQTD2016112514355531), the Science and Technology Innovation Committee Foundation of Shenzhen (Grant No. ZDSYS201703031748284), and the Program for University Key Laboratory of Guangdong Province (Grant No. 2017KSYS008).
Footnotes
 copyright: rightsretained
 doi: 10.1145/nnnnnnn.nnnnnnn
 isbn: 978xxxxxxxxxx/YY/MM
 journalyear: 2019
 copyright: acmcopyright
 conference: Genetic and Evolutionary Computation Conference; July 13–17, 2019; Prague, Czech Republic
 booktitle: Genetic and Evolutionary Computation Conference (GECCO ’19), July 13–17, 2019, Prague, Czech Republic
 price: 15.00
 doi: 10.1145/3321707.3321754
 isbn: 9781450361118/19/07
 ccs: Mathematics of computing Evolutionary algorithms
 Since SBX generates two children in a single operation, SBX is performed times.
 The monotonic improvement of the hypervolume value over time is guaranteed only when using the unbounded external archive (LópezIbáñez et al., 2011).
References
 Adaptation of expansion rate for realcoded crossovers. In GECCO, pp. 739–746. Cited by: 2nd item, §2.2, §3.
 Design of Evolutionary Computation for Continuous Optimization. Ph.D. Thesis, Tokyo Institute of Technology. Cited by: §2.2, §2.3, §2.3, §3, §4.2, §4.4.
 SMSEMOA: multiobjective selection based on dominated hypervolume. EJOR 181 (3), pp. 1653–1669. Cited by: §2.3.
 Generic Postprocessing via Subset Selection for Hypervolume and EpsilonIndicator. In PPSN, pp. 518–527. Cited by: §1, §1.
 Benchmarking Numerical Multiobjective Optimizers Revisited. In GECCO, pp. 639–646. Cited by: §1, §1, §3, §4.1.
 Biobjective Performance Assessment with the COCO Platform. CoRR abs/1605.01746. Cited by: §3, §4.1.
 The Pareto EnvelopeBased Selection Algorithm for Multiobjective Optimisation. In PPSN, pp. 839–848. Cited by: §1.
 Simulated Binary Crossover for Continuous Search Space. Complex Systems 9 (2). Cited by: §2.2, §2.2.
 A fast and elitist multiobjective genetic algorithm: NSGAII. IEEE TEVC 6 (2), pp. 182–197. Cited by: §1, §2.3.
 A Computationally Efficient Evolutionary Algorithm for RealParameter Optimization. Evol. Comput. 10 (4), pp. 345–369. Cited by: §2.2.
 Controlled Elitist Nondominated Sorting Genetic Algorithms for Better Convergence. In EMO, pp. 67–81. Cited by: §4.3.
 Evaluating the epsilonDomination Based MultiObjective Evolutionary Algorithm for a Quick Computation of ParetoOptimal Solutions. Evol. Comput. 13 (4), pp. 501–525. Cited by: §1.
 Multiobjective optimization using evolutionary algorithms. John Wiley & Sons. Cited by: §1.
 jMetal: A Java framework for multiobjective optimization. Adv. Eng. Softw. 42 (10), pp. 760–771. Cited by: §3.
 RealCoded Genetic Algorithms and IntervalSchemata. In FOGA, pp. 187–202. Cited by: §2.2.
 Genetic Algorithms for Multiobjective Optimization: FormulationDiscussion and Generalization. In ICGA, pp. 416–423. Cited by: §1, §1.
 COCO: A Platform for Comparing Continuous Optimizers in a BlackBox Setting. CoRR abs/1603.08785. Cited by: §3.
 RealParameter BlackBox Optimization Benchmarking 2009: Noiseless Functions Definitions. Technical report Technical Report RR6829, INRIA. Cited by: §3.
 Theoretical Analysis of Simplex Crossover for RealCoded Genetic Algorithms. In PPSN, pp. 365–374. Cited by: §2.2.
 Covariance Matrix Adaptation for Multiobjective Optimization. Evol. Comput. 15 (1), pp. 1–28. Cited by: §2.3, §5.
 A multiobjective genetic local search algorithm and its application to flowshop scheduling. IEEE Trans. SMC, Part C 28 (3), pp. 392–403. Cited by: §1.
 Optimization by simulated annealing. science 220 (4598), pp. 671–680. Cited by: §4.3.
 Theoretical Analysis of the Unimodal Normal Distiibution Crossover for Realcoded Genetic Algorithms. In IEEE CEC, pp. 529–534. Cited by: 2nd item, §2.2.
 Multiparental extension of the unimodal normal distribution crossover for realcoded genetic algorithms. In IEEE CEC, pp. 1581–1587. Cited by: §2.2.
 On Sequential Online Archiving of Objective Vectors. In EMO, pp. 46–60. Cited by: §1, §1, footnote 2.
 Crowding and Preselection Revisited. In PPSN, pp. 27–36. Cited by: §2.3.
 Nonlinear multiobjective optimization. Springer. Cited by: §1.
 A Real Coded Genetic Algorithm for Function Optimization Using Unimodal Normal Distributed Crossover. In GECCO, pp. 246–253. Cited by: §2.2.
 Multiple objective optimization with vector evaluated genetic algorithms. In ICGA, pp. 93–100. Cited by: §1.
 Distance based subset selection for benchmarking in evolutionary multi/manyobjective optimization. IEEE TEVC, pp. . Cited by: §1.
 Multiobjective Optimization Using Nondominated Sorting in Genetic Algorithms. Evol. Comput. 2 (3), pp. 221–248. Cited by: §1.
 Multiparent Recombination with Simplex Crossover in Real Coded Genetic Algorithms. In GECCO, pp. 657–664. Cited by: 2nd item, §2.2.
 COCO: The Biobjective Black Box Optimization Benchmarking (bbobbiobj) Test Suite. CoRR abs/1604.00359. Cited by: §1, §3.
 Performance of the DEMO Algorithm on the Biobjective BBOB Test Suite. In GECCO, pp. 1249–1256. Cited by: §3.
 Toward StepSize Adaptation in Evolutionary Multiobjective Optimization. In EMO, pp. 670–684. Cited by: §1, §1, §4.4, §5.
 Indicatorbased selection in multiobjective search. In PPSN, pp. 832–842. Cited by: §2.3.
 SPEA2: Improving the Strength Pareto Evolutionary Algorithm. Technical report ETHZ. Cited by: §1, §2.3.
 Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE TEVC 3 (4), pp. 257–271. Cited by: §1.