 # Matching-Based Selection with Incomplete Lists for Decomposition Multi-Objective Optimization

The balance between convergence and diversity is a key issue of evolutionary multi-objective optimization. The recently proposed stable matching-based selection provides a new perspective to handle this balance under the framework of decomposition multi-objective optimization. In particular, the stable matching between subproblems and solutions, which achieves an equilibrium between their mutual preferences, implicitly strikes a balance between the convergence and diversity. Nevertheless, the original stable matching model has a high risk of matching a solution with a unfavorable subproblem which finally leads to an imbalanced selection result. In this paper, we propose an adaptive two-level stable matching-based selection for decomposition multi-objective optimization. Specifically, borrowing the idea of stable matching with incomplete lists, we match each solution with one of its favorite subproblems by restricting the length of its preference list during the first-level stable matching. During the second-level stable matching, the remaining subproblems are thereafter matched with their favorite solutions according to the classic stable matching model. In particular, we develop an adaptive mechanism to automatically set the length of preference list for each solution according to its local competitiveness. The performance of our proposed method is validated and compared with several state-of-the-art evolutionary multi-objective optimization algorithms on 62 benchmark problem instances. Empirical results fully demonstrate the competitive performance of our proposed method on problems with complicated Pareto sets and those with more than three objectives.

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## 1 Introduction

The multi-objective optimization problem (MOP) considered in this paper is defined as follows :

 minimizeF(x)=(f1(x),⋯,fm(x))Tsubject tox∈Ω (1)

where is a

-dimensional decision vector and

is a -dimensional objective vector. is the feasible region of the decision space, while is the corresponding attainable set in the objective space . Given two solutions , is said to dominate , denoted by , if and only if for all and . A solution is said to be Pareto optimal if and only if no solution dominates it. All Pareto optimal solutions constitute the Pareto-optimal set (PS) and the corresponding Pareto-optimal front (PF) is defined as .

Evolutionary multi-objective optimization (EMO) algorithms, which are capable of approximating the PS and PF in a single run, have been widely accepted as a major approach for multi-objective optimization. Convergence and diversity are two cornerstones of multi-objective optimization: the former means the closeness to the PF while the latter indicates the spread and uniformity along the PF. Selection, which determines the survival of the fittest, plays a key role in balancing convergence and diversity. According to different selection mechanisms, the existing EMO algorithms can be roughly classified into three categories, i.e., Pareto-based methods

[2, 3, 4, LiDZZ16], indicator-based methods [5, 6, 7] and decomposition-based methods [8, 9, 10].

This paper focuses on the decomposition-based methods, especially the multi-objective evolutionary algorithm based on decomposition (MOEA/D)

. The original MOEA/D employs a steady-state selection mechanism, where the population is updated immediately after the generation of an offspring. In particular, this offspring is able to replace its neighboring parents when it has a better aggregation function value for the corresponding subproblem. To avoid a superior solution overwhelmingly occupying the whole population,  suggested to restrict the maximum number of replacements taken by an offspring. More recently,  developed a new perspective to understand the selection process of MOEA/D. Specifically, the selection process of MOEA/D is modeled as a one-one matching problem, where subproblems and solutions are treated as two sets of matching agents whose mutual preferences are defined as the convergence and diversity, respectively. Therefore, a stable matching between subproblems and solutions achieves an equilibrium between their mutual preferences, leading to a balance between convergence and diversity. However, as discussed in  and , partially due to the overrated convergence property, both the original MOEA/D and the stable matching-based selection mechanism fail to maintain the population diversity when solving problems with complicated properties, e.g., imbalanced problem [13, 14] and many objectives. Bearing these considerations in mind,  modified the mutual preference definition and developed a straightforward but more effective selection mechanism based on the interrelationship between subproblems and solutions. Later on,  proposed an adaptive replacement strategy, which adjusts the replacement neighborhood size dynamically, to assign solutions to their most suitable subproblems. It is also interesting to note that some works took the advantages of the Pareto dominance- and decomposition-based selection mechanisms in a single paradigm [LiKWCR12, 16, 17, 18, LiDZ15].

To achieve a good balance between convergence and diversity, this paper suggests to introduce the concept of incomplete preference lists into the stable matching model. Specifically, borrowing the idea from the stable matching with incomplete preference lists , we restrict the number of subproblems with which a solution is allowed to match. In this case, a solution can only be assigned to one of its favorite subproblems. However, due to the restriction on the preference list, the stable marriage model, which results in a one-one matching, may leave some subproblems unmatched. To remedy this situation, this paper implements two different versions of stable matching-based selection mechanisms with incomplete preference lists.

• The first one achieves a two-level one-one matching. At the first level, we find the stable solutions for subproblems according to the incomplete preference lists. Afterwards, at the second level, the remaining unmatched subproblems are matched with suitable solutions according to the remaining preference information.

• The second one obtains a many-one matching. In such a way, the unmatched subproblems give the matching opportunities to other subproblems that have already matched with a solution but still have openings.

Note that the length of the incomplete preference list has a significant impact on the performance and is problem dependent . By analyzing the underlying mechanism of the proposed stable matching-based selection mechanisms in depth, we develop an adaptive mechanism to set the length of the incomplete list for each solution on the fly. Comprehensive experiments on 62 benchmark problems fully demonstrate the effectiveness and competitiveness of our proposed methods.

The rest of the paper is organized as follows. sec:preliminaries introduces some preliminaries of this paper. Thereafter, the proposed algorithm is described step by step in sec:proposal. sec:setup and sec:experiments provide the experimental settings and the analysis of the empirical results. Finally, sec:conclusion concludes this paper and provides some future directions.

## 2 Preliminaries

In this section, we first introduce some background knowledge of MOEA/D and the stable matching-based selection. Then, our motivations are developed by analyzing their underlying mechanisms and drawbacks.

### 2.1 Moea/d

As a representative of the decomposition-based algorithms, MOEA/D has become an increasingly popular choice for posterior multi-objective optimization. Generally speaking, there are two basic components in MOEA/D: one is decomposition and the other is collaboration. The following paragraphs give some general descriptions of each component separately.

#### 2.1.1 Decomposition

The basic idea of decomposition is transforming the original MOP into a single-objective optimization subproblem. There are many established decomposition methods developed for classic multi-objective optimization , among which the most popular ones are weighted sum, Tchebycheff (TCH) and boundary intersection approaches. Without loss of generality, this paper considers the inverted TCH approach , which is defined as follows:

 (2)

where is a user specified weight vector, for all and . In practice, is set to be a very small number, say , when . is an Utopian objective vector where , . Note that the search direction of the inverted TCH approach is , and the optimal solution of (2) is a Pareto-optimal solution of the MOP defined in (1) under some mild conditions. We can expect to obtain various Pareto-optimal solutions by using (2

) with different weight vectors. In MOEA/D, a set of uniformly distributed weight vectors are sampled from a unit simplex.

#### 2.1.2 Collaboration

As discussed in , the neighboring subproblems, associated with the geometrically close weight vectors, tend to share similar optima. In other words, the optimal solution of is close to that of , given and are close to each other. In MOEA/D, each solution is associated with a subproblem. During the optimization process, the solutions cooperate with each other via a well-defined neighborhood structure and they solve the subproblems in a collaborative manner. In practice, the collaboration is implemented as a restriction on the mating and update procedures. More specifically, the mating parents are selected from neighboring subproblems and a newly generated offspring is only used to update its corresponding neighborhood. Furthermore, since different subproblems might have various difficulties, it is more reasonable to dynamically allocate the computational resources to different subproblems than treating all subproblems equally important. In , a dynamic resource allocation scheme is developed to allocate more computational resources to those promising ones according to their online performance.

### 2.2 Stable Matching-Based Selection

Stable marriage problem (SMP) was originally introduced in  and its related work won the 2012 Nobel Prize in Economics. In a nutshell, the SMP is about how to establish a stable one-one matching between two sets of agents, say men and women, which have mutual preferences over each other. A stable matching should not contain a man and a woman who are not matched together but prefer each other to their assigned spouses.

In MOEA/D, subproblems and solutions can be treated as two sets of agents which have mutual preferences over each other. In particular, a subproblem prefers a solution that optimizes its underlying single-objective optimization problem as much as possible; while a solution prefers to have a well distribution in the objective space. The ultimate goal of selection is to select the best solution for each subproblem, and vice versa. In this case, we can treat the selection procedure as a one-one matching procedure between subproblems and solutions. To the best of our knowledge, MOEA/D-STM  is the first one that has modeled the selection procedure of MOEA/D as an SMP, and encouraging results have been reported therein. The framework of the stable matching-based selection contains two basic components, i.e., preference settings and matching model. The following paragraphs briefly describe these two components.

#### 2.2.1 Preference Settings

The preference of a subproblem on a solution is defined as:

 ΔP(p,x)=gtch(x|w,z∗), (3)

where is the weight vector of . Consequently, measures the convergence of with respect to . The preference of a solution to a subproblem is defined as:

 ΔX(x,p)=∥¯¯¯¯F(x)−wT⋅¯¯¯¯F(x)wT⋅ww∥, (4)

where is the normalized objective vector of and is the -norm. Since the weight vectors are usually uniformly distributed, it is desirable that the optimal solution of each subproblem has the shortest perpendicular distance to its corresponding weight vector. For the sake of simplicity, can be used to measure the diversity of a solution .

#### 2.2.2 Matching Model

Based on the above preference settings,  employed the classic deferred acceptance procedure (DAP) developed in  to find a stable matching between subproblems and solutions. The pseudo code of this stable matching-based selection mechanism is given in alg:stm. and are the preference matrices of subproblems and solutions, each row of which represents the preference list of a subproblem over all solutions, and vice versa. In particular, a preference list is built by sorting the preference values in an ascending order. indicates the set of all the constructed matching pairs. It is worth noting that the convergence and diversity have been aggregated into the preference settings, thus the stable matching between subproblems and solutions strikes the balance between convergence and diversity.

### 2.3 Drawbacks of MOEA/D and MOEA/D-STM

In this subsection, we discuss some drawbacks of the selection mechanisms of MOEA/D and MOEA/D-STM.

#### 2.3.1 Moea/d

The update mechanism of the original MOEA/D is simple and efficient, yet greedy. In a nutshell, each subproblem simply selects its best solution according to the corresponding scalar optimization function value. As discussed in , since different parts of the PF might have various difficulties, some subproblems might be easier than the others for finding the optimal solutions. During some intermediate stages of the optimization process, the currently elite solutions of some relatively easier subproblems might also be good candidates for the others. In this case, these elite solutions can easily take over all subproblems. In addition, it is highly likely that the offspring solutions generated from these elite solutions crowd into the neighboring areas of the corresponding subproblems. Therefore, this purely fitness-driven selection mechanism can be severely harmful for the population diversity and may lead to the failure of MOEA/D on some challenging problems . Let us consider an example shown in fig:selection(a), where five out of ten solutions need to be selected for five subproblems. Since is currently the best solution for and is the current best candidate for , these two elite solutions finally take over all five subproblems. Obviously, the population diversity of this selection result is not satisfied.

#### 2.3.2 Moea/d-Stm

As discussed in , the DAP maximizes the satisfactions of the preferences of men and women in order to maintain the stable matching relationship. According to the preference settings for subproblems, solutions closer to the PF are always on the front of the subproblems’ preference lists. In this case, the DAP might make some solutions match themselves with subprobolems lying on the rear of their preference lists. Even worse, as discussed in sec:drawbacks_MOEAD, these currently well converged solutions may crowd in a narrow area. This obviously goes against the population diversity. Let us consider the same example discussed in fig:selection(a). The preference matrices of subproblems and solutions are:

 ΨP=p1:p2:p3:p4:p5:[ 12345678109 ] (5)
 ΨX=x1:x2:x3:x4:x5:x6:x7:x8:x9:x10:[ 12345 ] . (6)

From eq:pps, we can clearly see that to dominate the top positions of the preference lists of all subproblems. By using alg:stm, we have the selection/matching result shown in fig:selection(b), where to crowd in a narrow area between and . This is obviously harmful for the population diversity as well.

From the above discussions, we find that the original selection mechanism of MOEA/D is a convergence first and diversity second strategy , which might give excessive priority to the convergence requirement. On the other hand, although the stable matching-based selection mechanism intends to achieve an equilibrium between convergence and diversity, the stable matching between subproblems and solutions may fail to keep the population diversity. This is because no restriction has been given to the subproblem with which a solution can match. In other words, a solution can match with an unfavorable subproblem in the resulting stable matching. To relieve this side effect, the next section suggests a strategy to take advantages of some partial information from the preference lists when finding the stable matching between subproblems and solutions.

## 3 Adaptive Stable Matching-Based Selection with Incomplete Preference Lists

In the canonical SMP, each man/woman holds a complete and strictly ordered preference list over all agents from the other side. However, in practice, it may happen that a man/woman declares some unacceptable partners , and this results in an SMP with incomplete lists . By these means, a man/woman is only allowed to match with a matching agent that appears on his/her incomplete preference list. Due to the restriction from the incomplete preference lists, there is no guarantee that all agents can have a stable matching mate. A stable matching for an SMP with incomplete lists does not contain such a pair of man and woman: 1) they are acceptable to each other but not matched together; 2) they either do not match with anyone else or prefer each other to their current matching mates. To overcome the drawbacks discussed in sec:drawbacks, here we implement two versions of stable matching-based selection mechanisms with incomplete preference lists: one achieves a two-level one-one matching while the other obtains a many-one matching.

### 3.1 Two-Level One-One Stable Matching-Based Selection

In the first level, let us assume that there are subproblems and solutions, where . After obtaining the complete preference lists of all subproblems and solutions (line 1 and line 2 of alg:stm2l), we only keep the first , where and , subproblems on the preference list of each solution , while the remaining ones are not considered any longer (line 2 and line 3 of alg:stmic). In this case, each solution is only allowed to match with its first several favorite subproblems which are close to itself according to eq:preference_x. In contrast, the preference lists of subproblems are kept unchanged. Given the incomplete preference information, we employ the DAP to find a stable matching between subproblems and solutions (line 4 to line 12 of alg:stmic). By these means, we can expect that the population diversity is strengthened during the first-level stable matching. This is because a solution is not allowed to match with an unfavorable subproblem which lies out of its incomplete preference list. The pseudo code of the stable matching with incomplete lists is given in alg:stmic.

During the first-level stable matching, not all subproblems are assigned with a stable solution due to the incomplete preference information. To remedy this issue, the second-level stable matching with complete preference lists is developed to find a stable solution for each unmatched subproblem. At first, we compute the preference matrices of the unmatched subproblems and solutions (line 7 of alg:stm2l). Afterwards, we employ alg:stm to find a stable matching between them (line 8 of alg:stm2l). In the end, the matching pairs of both levels of stable matching are gathered together to form the final selection results (line 9 of alg:stm2l). The pseudo code of the two-level stable matching-based selection mechanism is given in alg:stm2l.

### 3.2 Many-One Stable Matching-Based Selection

Many-one stable matching problem is an extension of the standard SMP, where a matching agent from one side is allowed to have more than one matching mates from the other side. For example, in the college admission problem (CAP) , the colleges and applicants are two sets of matching agents. Each college has a preference list over all applicants and vice versa. Different from the SMP, each applicant is only allowed to enter one college, whereas each college has a positive integer quota being the maximum number of applicants that it can admit.

As the other implementation of the stable matching-based selection with incomplete preference lists, here we model the selection process of MOEA/D as a CAP with a common quota . More specifically, subproblems and solutions are treated as colleges and applicants respectively. A solution is only allowed to match with one subproblem while a subproblem is able to match with more than one solution. In particular, we do not limit the separate quota for every subproblem but assign a common quota for all subproblems, which equals the number of subproblems (i.e., ). In other words, subproblems can at most match with solutions in this many-one matching. Note that a matching is stable if there does not exist any pair of subproblem and solution where:

• and are acceptable to each other but not matched together;

• is unmatched or prefers to its assigned subproblem;

• the common quota is not met or prefers to at least one of its assigned solutions.

The pseudo code of the many-one stable matching-based selection mechanism is given in alg:mostmic. The initialization process (line 1 to line 5 of alg:mostmic) is the same as the one-one stable matching discussed in sec:twolevelstm. During the main while-loop, an unmatched solution at first matches with its current favorite subproblem according to its preference list (line 7 to line 11 of alg:mostmic). If the number of current matching pairs is larger than , we find a substitute subproblem and adjust its matching pairs by releasing the matching relationship with its least preferred solution (line 12 to line 18 of alg:mostmic). In particular, is selected according to the following criteria:

• At first, we choose the subproblems that have the largest number of matched solutions to form (line 13 of alg:mostmic). Its underlying motivation is to reduce the chance for overly exploiting a particular subproblem.

• If the cardinality of is greater than one, we need to further process . Specifically, we investigate the ranks of the solutions matched with subproblems in . The subproblems, whose least preferred solution holds the worst rank on that subproblem’s preference list, are used to reconstruct (line 14 of alg:mostmic).

• In the end, is randomly chosen from (line 14 of alg:mostmic).

Note that we add back into after releasing its matching relationship with (line 18 of alg:mostmic). The matching process terminates when becomes empty.

### 3.3 Impacts of the Length of the Incomplete Preference List

As discussed in the previous subsections, we expect to improve the population diversity by restricting the length of the preference list of each solution. A natural question is whether this length affects the behavior of our proposed stable matching-based selection mechanisms? Let us consider the example discussed in fig:selection again. For the sake of discussion, here we set the length of the incomplete preference list of each solution as a constant (denoted by ). By using different settings of , fig:stm2l-eg shows the selection results of the two-level stable matching-based selection mechanism. From this figure, we find that the diversity of the selected solutions increases with the decrease of ; on the other hand, the improvement of the diversity is at the expense of the convergence. It is interesting to note that the two-level stable matching-based selection mechanism totally degenerates into the original stable matching-based selection mechanism shown in fig:selection(a) when using . In a word, controls the trade-off between convergence and diversity in the stable matching-based selection with incomplete preference lists. In the next subsection, we develop an adaptive mechanism to control the length of each solution’s preference list on the fly.

To better understand the proposed adaptive mechanism, here we introduce the concept of local competitiveness. At first, all solutions are associated with their closest subproblems having the shortest perpendicular distance between the objective vector of the solution and the weight vector of the subproblem. Afterwards, for each subproblem having more than one associated solutions, we choose the one, which has the best aggregation function value, as its representative solution. A solution is defined as a locally competitive solution in case it dominates at least one representative solution of its nearest subproblems; otherwise, it is defined as a locally noncompetitive solution. In view of the population dynamics of the evolutionary process, we develop an adaptive mechanism to set the length of the incomplete preference list of a solution according to its local competitiveness (alg:r gives its pseudo code). Briefly speaking, this length is set as the maximum that keeps the corresponding solution locally noncompetitive.

More specifically, given subproblems and solutions, each solution is associated with its closest subproblem as shown in line 1 and line 2 of alg:r. In particular, represents the index of the subproblem with which a solution is associated, . In line 4 of alg:r, we collect the associated solutions of each subproblem , , to form a temporary set . Then, line 5 to line 8 of alg:r determine the representative solution of each subproblem , where represents the index of its representative solution. Afterwards, for each solution , line 10 to line 16 of alg:r gradually increase until becomes locally competitive, and this final is used as the length of ’s incomplete preference list. Note that since each solution locates within the subspace between closest neighboring weight vectors in -dimensional objective space, it can be associated with any of these subproblems in principle. Moreover, to avoid unnecessary comparisons, it is desirable to keep the solution’s incomplete preference list within a reasonably small length. All in all, the length of ’s incomplete preference list is adaptively tuned between and . In particular, is set as the neighborhood size used in MOEA/D, where the mating parents are selected from.

Let us use the example shown in fig:selection(b) to explain the underlying principle of our proposed adaptive mechanism. In this example, solutions and become locally competitive when ; while solutions and are locally noncompetitive for all settings. It is worth noting that neither nor is the representative solution of any subproblem; in the meanwhile, they are crowded in a narrow area. Since these locally competitive solutions have better ranks in the preference lists than those less competitive ones, the original stable matching-based selection tends to give them higher priorities to form the matching pairs. However, this selection result is obviously harmful for the population diversity. In addition, we also notice that and are the representative solutions of and , thus they should contain some relevant information for optimizing these subproblems. In contrast, although and have better aggregation function values, they are far away from and and should be less relevant to them. To resolve these issues, our proposed adaptive mechanism adaptively restricts the length of the preference list of each solution by removing subproblems whose representative solution is dominated by . By these means, we can make sure that each solution does not consider a subproblem which prefers this solution to its own representative solution. Thus each subproblem is prevented from matching with a less relevant solution. Note that this adaptive mechanism can be readily plugged into both of our proposed two versions of stable matching-based selection mechanisms by using alg:r to replace line 2 of alg:stm2l and alg:mostmic, respectively. The adaptive two-level one-one stable matching-based selection mechanism and the adaptive many-one stable matching-based selection mechanism are denoted by AOOSTM and AMOSTM for short.

### 3.5 Time Complexity of AOOSTM and AMOSTM

In this subsection, we analyze the complexity of AOOSTM and AMOSTM. For both selection mechanisms, the calculation of and cost computations . In alg:r, the association operation between subproblems and solutions costs calculations (line 1 to line 2). As for line 3 to line 8 of alg:r, the identification of the representative solution for each subproblem requires computations. Thereafter, the computation of in line 9 to line 16 of alg:r costs computations in the worst case. Considering the two-level one-one stable matching in alg:stm2l, the complexity of the one-one stable matching with the incomplete lists in line 3 is , which is simpler than the original stable matching with complete preference lists . Next, the complexity of line 4 to line 6 of alg:stm2l is . During the second-level stable matching (line 7 to line 8 of alg:stm2l), same complexity analysis can be done for the remaining subproblems and solutions. Overall, the total complexity of AOOSTM is . When it comes to AMOSTM, since alg:mostmic is solution-oriented, the computational complexity of line 3 to line 19 is . The total complexity of AMOSTM is still .

### 3.6 Incorporation with MOEA/D

Similar to , we choose the MOEA/D-DRA  as the base framework and replace the update mechanism by the AOOSTM and AMOSTM selection mechanisms developed in sec:adaptive. The resulted algorithms are denoted by MOEA/D-AOOSTM and MOEA/D-AMOSTM, of which the pseudo code is given in alg:moead-astm. Note that the normalization scheme proposed in  is adopted to handle MOPs with different scales of objectives. In the following paragraphs, some important components of MOEA/D-AOOSTM/AMOSTM are further illustrated.

#### 3.6.1 Initialization

Without any prior knowledge of the landscape, the initial population is randomly sampled from . Same as the original MOEA/D, we use the classic method suggested in  to generate a set of uniformly distributed weight vectors on a unit simplex. In addition, for each weight vector , , we assign its , , closest weight vectors as its neighbors.

#### 3.6.2 Reproduction

According to the underlying test problem, here we employ the widely used differential evolution (DE) , simulated binary crossover (SBX)  and polynomial mutation  for offspring generation.

#### 3.6.3 Utility of Subproblem 

The utility of subproblem , denoted by , , measures the improvement rate of . We make some modifications on to fit our proposed MOEA/D-AOOSTM/AMOSTM:

 πi=⎧⎪⎨⎪⎩1if Δi>0.0010.95×πiif Δi<0(0.95+0.05×Δi0.001)×πiotherwise (7)

where represents the relative decrease of the scalar objective value of and is evaluated as:

 Δi=⎧⎨⎩gtch(xi,old|wi,z∗)−gtch(xi,best|wi,z∗)gtch(xi,old|wi,z∗)if M(pi)≠∅0otherwise (8)

where is the best solution matched with in the current generation and is the previously saved value of .

## 4 Experimental Settings

This section presents the general setup of our empirical studies, including the benchmark problems, algorithms in comparisons, parameter settings and performance metrics.

### 4.1 Benchmark Problems

From three popular benchmark suites, i.e., MOP , UF  and WFG , 62 problem instances in total, are chosen as the benchmark set in our empirical studies. These problem instances have various characteristics, e.g., non-convexity, deceptive, multi-modality. According to the recommendations in the original references, the number of decision variables is set as: for the MOP instances and for the UF instances. As the WFG instances are scalable to any number of objectives, here we consider . In particular, when ,  , where the position-related variable and the distance-related variable ; while for , we use the recommended settings in  and , i.e., and .

### 4.2 Algorithms in Comparisons

Nine state-of-the-art EMO algorithms, i.e., MOEA/D-STM, MOEA/D-IR , gMOEA/D-AGR , MOEA/D-M2M , MOEA/D-DRA, HypE , NSGA-III , PICEA-g  and MOEA/DD , are considered in our empirical studies. In particular, the first seven algorithms are used for comparative studies on problems with complicated PSs; while the latter five are chosen to investigate the scalability on problems with more than three objectives. The characteristics of these algorithms are briefly described in the supplementary file of this paper.

### 4.3 Parameter Settings

Referring to [10, 12] and , the settings of the population size for different benchmark problems are shown in tab:popsize. The stopping condition of each algorithm is the predefined number of function evaluations. In particular, it is set to for the UF and MOP instances , and for the bi-objective WFG instances . As for the many-objective WFG instances, where , the number of function evaluations is set as , , and , respectively . The parameters of our proposed MOEA/D-AOOSTM and MOEA/D-AMOSTM are set as follows:

• Reproduction operators: As for problems with complicated properties, we use the DE operator and polynomial mutation for offspring generation. As recommended in , we set and for the UF and MOP instances; while

for bi-objective WFG instances. The mutation probability

is set to be and its distribution index equals . For problems with more than three objectives, we use the SBX operator to replace the DE operator, where the crossover probability and its distribution index  . All other MOEA/D variants in our experimental studies share the same settings for reproduction operators.

• Neighborhood size:  [10, 12].

• Probability to select in the neighborhood:  .

### 4.4 Performance Metrics

To assess the performance of different algorithms, we choose the following two widely used performance metrics:

1. Inverted Generational Distance (IGD) : Given as a set of points uniformly sampled along the PF and as the set of solutions obtained from an EMO algorithm. The IGD value of is calculated as:

 IGD(P,P∗)=∑z∈P∗dist(z,P)|P∗|, (9)

where is the Euclidean distance of to its nearest point in .

2. Hypervolume (HV) : Let be a point dominated by all the Pareto optimal objective vectors. The HV of is defined as the volume of the objective space dominated by the solutions in and bounded by :

 HV(P)=VOL(⋃z∈P[z1,zr1]×⋯×[zm,zrm]), (10)

where VOL indicates the Lebesgue measure.

Since the objective functions of WFG instances are in different scales, we normalize their PFs and the obtained solutions in the range of before calculating the performance metrics. In this case, we constantly set

in the HV calculation. Note that both IGD and HV can evaluate the convergence and diversity simultaneously. A smaller IGD value or a large HV value indicates a better approximation to the PF. Each algorithm is independently run 51 times. The mean and standard deviation of the IGD and HV values are presented in the corresponding tables, where the ranks of each algorithms on each problems are also given by sorting the mean metric values. The best metric values are highlighted in boldface with a gray background. To have a statistically sound conclusion, we use the Wilcoxon’ s rank sum test at a significant level of 5% to evaluate whether the proposed MOEA/D-AOOSTM and MOEA/D-AMOSTM are significantly better or worse than the others. In addition, we use the two-sample Kolmogorov-Smirnov test at a significant level of 5% to summarize the relative performance of all test EMO algorithms.

## 5 Empirical Studies

In this section, we first analyze the comparative results for problems with complicated properties. Afterwards, we investigate the effectiveness of the adaptive mechanism. In the end, we summarize the experimental studies in a statistical point of view. Due to the page limits, the empirical studies on problems with more than three objectives are given in the supplementary file of this paper.

### 5.1 Performance Comparisons on MOP Instances

As discussed in 

, MOP benchmark suite, in which different parts of the PF have various difficulties, poses significant challenges for maintaining the population diversity. tab:mop-igd and tab:mop-hv demonstrate the IGD and HV results of the nine EMO algorithms. From the IGD results shown in tab:mop-igd, it can be seen that MOEA/D-AMOSTM shows the best overall performance and MOEA/D-AOOSTM, obtaining a slightly lower total rank than MOEA/D-M2M, ranks in the third place. In terms of the mean IGD values, MOEA/D-M2M gives the best results on MOP1 to MOP4, while MOEA/D-AOOSTM ranks the first on MOP5 and MOEA/D-AMOSTM beats all other EMO algorithms on MOP6 and MOP7. When it comes to the Wilcoxon’s rank sum test results, both MOEA/D-AMOSTM and MOEA/D-AOOSTM are significantly better the others on MOP2, MOP3 and MOP5 to MOP7. They are only beaten by MOEA/D-M2M on MOP1 and MOP4. This is because MOEA/D-AMOSTM and MOEA/D-AOOSTM achieve better performance on MOP2 and MOP3 than MOEA/D-M2M but the former two have large variances. Comparing MOEA/D-AOOSTM and MOEA/D-AMOSTM, they have no significant differences on five problems but the former is outperformed by the latter on MOP4 and MOP7. Following the best three algorithms, MOEA/D-IR and gMOEA/D-AGR are able to obtain a set of non-dominated solutions moderately covering the entire PF. As for MOEA/D-DRA, MOEA/D-STM, NSGA-III and HypE, they can only obtain some solutions lying on the boundaries. tab:mop-hv shows similar results in HV tests, except that MOEA/D-AMOSTM obtains better performance than MOEA/D-AOOSTM on MOP5.

We plot the final solution sets with the best IGD values in 51 runs on all test instances in the supplementary file of this paper. From Fig. 1 to Fig. 4 of the supplementary file, we can see that although MOEA/D-M2M obtains slightly better mean IGD and HV metric values than MOEA/D-AMOSTM and MOEA/D-AOOSTM, the solutions obtained by MOEA/D-AMOSTM and MOEA/D-AOOSTM have a more uniform distribution along the PF. This can be explained by the density estimation method, i.e., the crowding distance of NSGA-II, used in MOEA/D-M2M, which is too coarse to guarantee the population diversity. Nevertheless, the convergence ability of MOEA/D-M2M is satisfied, thus contributing to promising IGD values on MOP1 to MOP4. According to



, gMOEA/D-AGR uses a sigmoid function to assign a same replacement neighborhood size to all subproblems. However, since different parts of the PF require various efforts, this same setting might not be appropriate for all subproblems. From Fig. 3 and Fig. 7 of the supplementary file, we can obverse that the solutions obtained by gMOEA/D-AGR may miss some segments of the PF. This can be explained by the replacement neighborhood that grows too fast for the corresponding subproblems. In order to emphasize the population diversity, for each subproblem, MOEA/D-IR selects the appropriate solution from a couple of related ones. However, its preference setting, which encourages the selection in a less crowded area, tends to result in an unstable selection result. In this case, some solutions far away from the PF can be selected occasionally. The reason behind the poor performance of NSGA-III, HypE and MOEA/D-DRA is that their convergence first and diversity second selection strategies may easily trap the population in some narrow areas. As discussed in sec:drawbacks, the stable matching model used in MOEA/D-STM can easily match a solution with an unfavorable subproblem, thus resulting in an unbalanced selection.

### 5.2 Performance Comparisons on UF Instances

The comparison results on the IGD and HV metrics between MOEA/D-AOOSTM, MOEA/D-AMOSTM and the other EMO algorithms on UF benchmark suite are presented in tab:uf-igd and tab:uf-hv. Different from the MOP benchmark suite, the major source of difficulty for the UF benchmark suite is not the diversity preservation but the complicated PS. Generally speaking, the overall performance of MOEA/D-AOOSTM ranks the first on the UF benchmark suite, followed by MOEA/D-AMOSTM and their predecessor MOEA/D-STM. More specifically, for both the IGD and HV metrics, MOEA/D-AOOSTM performs the best on UF1, UF2 and UF8 and acts as the top three algorithm on all instances except for UF4, UF5 and UF10. For UF3 and UF7, the performance of MOEA/D-AOOSTM does not show significant difference with the best performing algorithms. MOEA/D-AMOSTM shows similar rankings to MOEA/D-AOOSTM. It is significantly better than MOEA/D-AOOSTM on UF4-UF6 and UF9 in terms of both the IGD and HV metrics. In contrast, MOEA/D-AOOSTM wins on UF2, UF8 and UF10 according to Wilcoxon’s rank sum test of the IGD results and wins on UF1, UF2, UF7, UF8 and UF10 in the HV tests.

According to the performance of different algorithms on the UF test instances, the analysis can be divided into three groups. For UF4 and UF5, MOEA/D-M2M, NSGA-III and HypE are able to provide better performance than all other MOEA/D variants. All these three algorithms use the Pareto dominance as the major driving force in the environmental selection, which can improve the convergence to a great extent. For UF1 to UF3 and UF6 to UF9, all the MOEA/D variants outperform NSGA-III and HypE. In particular, The three variants with stable matching-based selection, i.e., MOEA/D-AOOSTM, MOEA/D-AMOSTM and MOEA/D-STM, have shown very promising results on these six test instances. The superior performance can be attributed to the well balance between convergence and diversity achieved by the stable matching relationship between subproblems and solutions. gMOEA/D-AGR has shown a medium performance for the former two groups of problem instances. This might be due to its adaptive mechanism that can hardly make a satisfied prediction of the replacement neighborhood size. UF10 is a difficult tri-objective problem, where none of these eight EMO algorithms are able to obtain a well approximation to the PF within the given number of function evaluations. Nevertheless, it is worth noting that the empirical studies in  demonstrate that the stable matching-based selection mechanism can offer a competitive result in case the maximum number of function evaluations is doubled.

### 5.3 Performance Comparisons on Bi-Objective WFG Instances

From the comparison results shown in Table I and Table II of the supplementary file, it can be seen that MOEA/D-AOOSTM and MOEA/D-AMOSTM are the best two algorithms in overall performance on the bi-objective WFG instances. Comparing with the seven existing algorithms, MOEA/D-AOOSTM and MOEA/D-AMOSTM achieves significant better performance in 56 and 57 out of 63 IGD comparisons respectively. As for HV results, they both wins in 57 comparisons. In particular, MOEA/D-AOOSTM and MOEA/D-AMOSTM obtain the best mean metric values on WFG1, WFG3, WFG6, WFG7 and WFG9 and obtain very promising results on WFG3 and WFG5. Even though the mean HV metric values of MOEA/D-AOOSTM and MOEA/D-AMOSTM on WFG2 rank the fifth and sixth, all the other algorithms are significantly worse than them. It is interesting to note that MOEA/D-AOOSTM and MOEA/D-AMOSTM are significantly better than MOEA/D-STM on all WFG instances except for WFG8. One possible reason is the proper normalization method used in MOEA/D-AOOSTM/AMOSTM. However, we also notice that the performance of NSGA-III fluctuates significantly on difficult problem instances, though it uses the same normalization method. The other variants of MOEA/D perform more or less the same on all test instances except MOEA/D-M2M significantly outperforms all other EMO algorithms on WFG8. The indicator-based algorithm HypE performs the worst on all nine problems. Comparing the IGD and HV results, the algorithm comparisons are consistent on most test instances, except when the performance of two algorithms are very close, the ranking of IGD and HV metric values may change slightly. However, it is worth noting that the algorithms perform quite differently on WFG2 under the IGD and HV assessments. NSGA-III shows the best mean IGD value but gives the second worst mean HV value. In contrast, the mean HV value of gMOEA/D-AGR ranks the first among all algorithms but its mean IGD value only obtains a rank of . This is probably because WFG2 has a discontinuous PF, which makes the distinction between IGD and HV more obvious.