## Abstract

There can be a complicated mapping relation between decision variables and objective functions in multi-objective optimization problems (MOPs). It is uncommon that decision variables influence objective functions equally. Decision variables act differently in different objective functions. Hence, often, the mapping relation is unbalanced, which causes some redundancy during the search in a decision space. In response to this scenario, we propose a novel memetic (multi-objective) optimization strategy based on dimension reduction in decision space (DRMOS). DRMOS firstly analyzes the mapping relation between decision variables and objective functions. Then, it reduces the dimension of the search space by dividing the decision space into several subspaces according to the obtained relation. Finally, it improves the population by the memetic local search strategies in these decision subspaces separately. Further, DRMOS has good portability to other multi-objective evolutionary algorithms (MOEAs); that is, it is easily compatible with existing MOEAs. In order to evaluate its performance, we embed DRMOS in several state of the art MOEAs to facilitate our experiments. The results show that DRMOS has the advantage in terms of convergence speed, diversity maintenance, and portability when solving MOPs with an unbalanced mapping relation between decision variables and objective functions.

## 1 Introduction

Many excellent MOEAs have been developed recently, and some of them have been summarized in the literature (Coello, 1999; Zitzler and Thiele, 1998; Zitzler et al., 2000; Khare et al., 2003). Among them, NSGA-II (Deb et al., 2002a) and SPEA2 (Zitzler et al., 2001) are the most popular. NSGA-II is well known for its fast non-dominated sort on dominance ranking and crowding distance on diversity maintenance, and SPEA2 is well known for its environmental selection on the basis of both dominance ranking and diversity maintenance. Recently, some MOEAs with new selection techniques have been proposed. For example, -MOEA (Deb et al., 2005) adopts -dominance to improve its performance; MOEA/D (Zhang et al., 2008b) applies the decomposition idea as a new selection pressure; hypervolume-based MOEA (Bader and Zitzler, 2011) performs well on many-objective optimization problems; TDEA (Karahan and Koksalan, 2010) uses the territory defining idea in its diversity maintenance mechanism; and memetic-based MOEAs (Goh et al., 2009; Knowles and Corne, 2000) use local search to solve MOPs.

As the mapping relation between the decision variables and the objective functions of MOPs is considerably more complicated than that of single-objective optimization problems, MOEAs face new challenges. It is quite uncommon for an MOP that all its decision variables influence one objective function value to the same extent. An unbalanced mapping relation in which different decision variables affect any given objective value differently often appears. However, existing MOEAs treat all the decision variables equally. Because of this, they waste their search resources on decision variables that only slightly affect one objective function. Thus, the unbalanced mapping relation causes some redundancy as a result of unnecessary search. MOEAs should focus their search resources on the decision variables that significantly affect the objective function, which is a type of dimension reduction. The mapping relation is a natural characteristic of MOPs, which can be regarded as prior knowledge. MOEAs using prior knowledge for local search can be seen as memetic algorithms (MAs). In order to reduce the difficulty of MOPs, a considerable amount of work on dimension reduction (Brockhoff and Zitzler, 2009; Saxena and Deb, 2007; López Jaimes et al., 2009; Corne and Knowles, 2007) and prior knowledge-based MAs (Meuth et al., 2009; Ong et al., 2010) has already been carried out. A brief introduction of these two types of technologies is presented below.

### Dimension Reduction

Dimension reduction has been widely applied in the fields of data mining and statistics as it maximally approximates the original problems by keeping important features and deleting others. Thus, the difficulties of the original problems can be reduced. In the field of many-objective optimization problems (MOPs with more than three objectives), dominance relations (Kukkonen and Lampinen, 2007; Sato et al., 2007), preference-based methods (Thiele et al., 2009), visualization (Pryke et al., 2007) and dimension reduction (Brockhoff and Zitzler, 2009) are the four main topics. The study of dimension reduction focuses mainly on the objective space. As the Pareto dominance relation hardly contributes to the selection in many-objective optimization problems, it is necessary to reduce the dimension of the objective space. The dimension can be reduced when objectives are correlated (Ishibuchi et al., 2011). The mainstream methods can be classified into three types: dimension reduction preserving the dominance relation (Brockhoff and Zitzler, 2009), dimension reduction based on a feature selection technique (López Jaimes et al., 2008), and dimension reduction by PCA to remove the less important objectives (Jolliffe, 2002). In the decision space of an MOP, the linkage between variables also increases its difficulty (Deb et al., 2006). Some researchers have attempted to reduce the dimension of decision space. For example, RM-MEDA (Zhang et al., 2008a) approximates a solution set by several lines.

### MA

MA, emphasizing heuristic search is attracting increasing attention. Unlike the global search evolutionary algorithm (EA), MA is a combination of heuristic local search and global search (Meuth et al., 2009; Ong et al., 2010). In MA, a meme represents a kind of local search. Lamarckian learning (Le et al., 2009; Ong and Keane, 2004; Liang et al., 2000a, 2000b), the multi-meme MA (Krasnogor and Smith, 2005), and Baldwinian learning (Gong et al., 2010) are different types of MAs. Recently, the adaptive MA, which adaptively selects suitable memes for different problems, has become popular in the field of MAs (Ong et al., 2006). MAs are powerful in solving real-world problems with prior knowledge. Memes can be designed according to the prior knowledge. MAs aim to solve specific problems, such as traveling salesman problems (TSPs; Lim et al., 2008), job-shop scheduling problems (Hasan et al., 2009), filter design (Tirronen et al., 2008), HIV multidrug therapy design (Neri et al., 2007), and PMSM drives control design (Caponio et al., 2007). It is worth noting that MA has already been successfully applied to MOPs (Goh et al., 2009; Knowles and Corne, 2000).

Taking the related work mentioned above as a foundation, we propose a memetic optimization strategy based on dimension reduction in decision space (DRMOS) in this paper. This strategy improves individuals by local search in the decision subspaces after dimension reduction. Given below are the contributions of this paper.

### Analysis of the Relation Between Decision Variables and Objective Functions

An unbalanced mapping relation between decision variables and objective functions is very common in most MOPs, which is obtained as the prior knowledge for memetic local search strategies. DRMOS obtains the relation between decision variables and objective functions by a statistical analysis approach from samples.

### Memetic Local Search Strategy

With the above heuristic information about the relation between decision variables and objective functions, DRMOS divides the decision space into several subspaces for memetic local search strategies, which decreases the dimension of the decision space.

### Portability to Other MOEAs

DRMOS is similar to a patch in the system of MOEAs which can improve their performance on MOPs with the unbalanced mapping relation.

The rest of this paper is organized as follows. The related definitions are introduced in Section 2. Section 3 describes the details of DRMOS, such as its basic idea, a relation analysis approach, memetic local search strategies, and its portability to MOEAs. Comparative experiments with the applications of DRMOS to other MOEAs are presented in Section 4. Finally, Section 5 concludes the discussion.

## 2 Related Definitions

### 2.1 Definitions of MOP

*m*-objective optimization problem can be represented as Equation (1), where is its feasible space and is the decision variable. is the objective function in an

*R*space.

^{m}If and are two decision vectors, then their corresponding objective values are and . If , and , then it is denoted as . is *Pareto optimal*, only if there is no such that . The set of all *Pareto optimal* solutions in *X* is called the *Pareto Set (PS)*. Further, the set of all the *Pareto optimal* objective values is called the *Pareto Front (PF*; Miettinen, 1999).

### 2.2 Decision Subspace

*x*

_{3}is unrelated to objective

*f*

_{1}. The search for

*x*

_{3}is unnecessary for

*f*

_{1}. However, the case in Equation (2) is an extreme situation of the unbalanced mapping relation. Mostly, the unbalanced mapping relation appears in MOPs with some decision variables that have little influence on some objective functions. In summary, redundancy does exist in the search space of some objectives. The decision space of MOPs with the unbalanced mapping relation can be divided into several subspaces for dimension reduction.

*m*objectives, its decision space can be divided into

*m*+ 1 decision subspaces. Subspace

*S*() is the subspace spanned by the decision variables related to objective

_{i}*f*only. Equation (3) is its definition, where is the orthogonal basis of subspace

_{i}*S*,

_{i}*J*is the index set of . is the subspace spanned by decision variables . It is noteworthy that the partial derivatives in Equation (3) simply represent whether the decision variables and the objective functions are related (including both linear and nonlinear correlation). There are no requirements for the differentiability of the objective functions. The

_{i}*m*+ 1th decision subspace

*S*

_{others}is the orthogonal complementary set of the former

*m*subspaces’ union set, which is shown in Equation (4). Subspace

*S*

_{others}is spanned by the decision variables related to multiple objectives. As these

*m*+ 1 subspaces are definitely disjointed, their direct sum is the entire decision space as shown in Equation (5), where “+” means sum, and “” means direct sum. The distance

*d*in subspace

*S*can be calculated using Equation (6), in which

*J*is the index set of

*S*’s orthogonal basis and

*d*is the projection on

_{j}*x*.

_{j}It is clear that objective *f _{i}* can be optimized in subspace

*S*independently, with no influence on the other objectives. It is obvious that subspace

_{i}*S*is smaller than the entire search space. The dimension of the decision space is reduced through the decision subspace division. For an individual

_{i}*P*, objective

*f*can be optimized independently by simply searching in subspace

_{i}*S*. Then, the obtained individual cannot be worse than

_{i}*P*. In other words, subspaces

*S*() only affect convergence. Further, the decision variables in subspace

_{i}*S*

_{others}are related to multiple objectives. Subspace

*S*

_{others}is related to both convergence and diversity. If every objective function depends on all the decision variables in an MOP, it cannot be decomposed into such subspaces by the above strong assumption. It is still an unbalanced case if decision variables act differently for different objective functions. It is less progressive to search on the decision variables that influence objective functions slightly. The above concept is expanded for common MOPs in Section 2.3.

### 2.3 MOPs Can Be Dimension-Reduced in Decision Space

#### 2.3.1 MOPs Can Be Strictly Dimension-Reduced in Decision Space

*n*is the dimension of the entire decision space. If an MOP strictly satisfies the above definition, it can be referred to as strictly dimension-reduced in the decision space.

#### 2.3.2 MOPs Can Be Weakly Dimension-Reduced in Decision Space

The MOPs that can be strictly dimension-reduced in the decision space are very special because the requirements of the decision subspace division are very strict. Without losing generality, the requirements of strict dimension reduction should be relaxed for the weak dimension reduction. Subspace *S _{i}* () is redefined as the subspace spanned by the decision variables that considerably impact objective

*f*only. If an MOP satisfies Equation (7) in its redefined subspace, it has an unbalanced mapping relation between decision variables and objective function values and it can be weakly dimension-reduced in the decision space. Furthermore, the relation between the decision variables and the objective functions can be expressed in terms of some statistical features. Our specific approach is discussed in Section 3.2.

_{i}If the variance of one objective is large, it means that the decision variable has a significant influence on that objective function; otherwise, it suggests that the influence of the decision variable on that objective function is not significant.

Provided that an MOP can be weakly dimension-reduced in its decision space, the median of objectives’ variances can be set as the threshold for the measurement of the influence between one decision variable and objectives. If one objective’s variance is larger than the threshold, the decision variable is regarded to have considerable influence on the objective.

#### 2.3.3 Reduction Rate

*S*

_{others}, there are multiple objectives to be considered; consequently, their weights in the reduction rate are larger than the ones in subspaces

*S*().

_{i}## 3 Memetic Optimization Strategy Based on Dimension Reduction in Decision Space

### 3.1 Basic Idea

All MOPs are considered as the problems that can be weakly dimension-reduced in DRMOS. DRMOS aims at improving the searching ability of existing MOEAs on such MOPs. DRMOS obtains the mapping relation to reduce the dimension of the decision space and applies memetic local search strategies in the divided decision subspaces; the flow-chart of DRMOS is shown in Figure 1.

As shown in Figure 1, DRMOS consists of two major procedures, namely, relation analysis and memetic local search. The former is used for gaining information where the mapping relation is learned via sampling. Then the decision space is divided into several subspaces according to that heuristic information. Finally, memetic local search strategies are applied to optimize each objective in their corresponding decision subspaces. The relation analysis result is dynamically updated. Memetic local search strategies are adjusted according to the dynamical information updated by the relation analysis approach.

### 3.2 Relation Analysis Approach

The mapping relation plays an important role in DRMOS. Some work on the prediction of such a relation has been done. For example, an artificial neural network (Adra et al., 2009; Gaspar-Cunha and Vieira, 2004) is used for mapping an objective space locally back to the decision space; the estimation of distribution algorithm (EDA) in Larranaga and Lozano (2002) builds a probability distribution model of variables on the basis of the statistical information; a Bayesian network (Laumanns and Ocenasek, 2002; Khan et al., 2002) adopts a probabilistic model of variables. The mentioned work serves as an inspiration to form our relation analysis approach.

*C*. Multiple samplings and predictions are adopted in our approach. Their consistency increases

*C*, while their inconsistency decreases

*C*. Our approach is described in Table 2 in detail by MATLAB notations. In Table 2, is the credibility of the prediction between decision variable

*x*and objective

_{j}*f*after the

_{k}*i*th sampling. When it is sufficiently high, the prediction can be used for the later memetic local search strategies. Because the mapping relation of

*x*to all the objectives in MOPs is independent, the total credibility can be calculated using Equation (9). Further, is used as a kind of probability to control the sampling on

_{j}*x*. That is, when is large, too many times of sampling on

_{j}*x*are unnecessary.

_{j}As indicated above, the obtained mapping relation is a kind of prediction result, which means that it may not be the right result. When all the credibility of decision variable *x _{j}* () is larger than a threshold

*T*, the relation prediction can be used for the decision subspace division. The process of relation analysis is very important for DRMOS. Without the mapping relation, memetic local search strategies cannot be applied.

*T*plays a very important role in DRMOS. If

*T*is small, then the prediction may be a wrong guide for memetic local search strategies in the divided subspace. If

*T*is large, it would incur a significant computational cost for sampling. The experimental analysis of the influence of

*T*on the entire DRMOS is discussed in Section 4.2.

### 3.3 Memetic Local Search Strategy

According to the mapping relation obtained in the relation analysis approach, the entire decision space can be divided into several disconnected subspaces for the separate optimization. That is, objective *f _{i}* can be optimized independently through a local search in subspace

*S*, where the decision variables with little influence on other objectives can be ignored. The memetic local search strategies in DRMOS aim at improving individuals through the search in subspace

_{i}*S*() to optimize objective

_{i}*f*as shown in Table 3. Every objective is optimized in the corresponding subspace, which is relatively easy to be solved even by a classical genetic algorithm (GA) in DRMOS. In the experiments referred to in this paper, a classical GA is used for the local search. The stopping criterion is 2000 function evaluations.

_{i}#### 3.3.1 Two Memetic Local Search Strategies

Convergence and diversity are both important for MOEAs. Two memetic local search strategies are designed to improve the performance on convergence and diversity in DRMOS. They both improve the individuals in subspace *S _{i}* () as shown in Table 3; however, they improve different individuals. Strategy 1 aims to improve the individuals in the current population to improve convergence. Strategy 2 aims at diversity. Unlike Strategy 1, Strategy 2 does not choose the individuals in the current population. It adds artificial individuals in the less-explored areas of subspace

*S*

_{others}, where the individuals in the current population are not crowded by the harmonic distance measurement (Wang et al., 2010). As Table 4 shows, the center of the neighborhood of the individual with the largest harmonic distance is used as the artificial individual in Strategy 2.

Parameter: m: the number of objectives | |

1 | Calculate the m-neighbor harmonic distances in S_{other} of all the solutions. |

2 | Find the individual P with the largest harmonic distance. |

3 | Artificial individual P is the center of _{a}P’s m-nearest neighbors. |

4 | Local search P in _{a}S () as in Table 3. _{i} |

Parameter: m: the number of objectives | |

1 | Calculate the m-neighbor harmonic distances in S_{other} of all the solutions. |

2 | Find the individual P with the largest harmonic distance. |

3 | Artificial individual P is the center of _{a}P’s m-nearest neighbors. |

4 | Local search P in _{a}S () as in Table 3. _{i} |

A memetic local search strategy requires extra function evaluations. Therefore, memetic local search strategies in DRMOS should be made good use of, but waste function evaluations if they are not properly used. For all MAs, the balance between global search and local search is an important research problem (Ishibuchi and Murata, 1998; Ishibuchi et al., 2003; Jaszkiewicz, 2002). In DRMOS, when the results of the relation analysis approach are reliable (all the values are larger than *T*), memetic local search strategies can be executed. In order to reduce the number of function evaluations, DRMOS avoids improving the similar individuals in memetic local search strategies. DRMOS opens a set *record* to store the improved individuals. After an individual is improved by a memetic local search, its decision variables are copied into *record* as a reference for the similarity measurement for the other individuals. In other words, before applying memetic local search strategies to a selected individual *P*, a similarity comparison with all the individuals in *record* is carried out, where the similarity is measured by the Euclidean distance in *S*_{others}. If the distance is smaller than the diagonal of the feasible area in *S*_{others}, DRMOS will drop this individual and choose another one from the current population until the entire population is compared.

#### 3.3.2 Interaction between Two Strategies

Convergence and diversity are two important topics in MOPs. However, when the computational time is limited, convergence must be considered first. In DRMOS, Strategy 1 is employed first to improve convergence. When all the individuals in the current population are similar to the members in *record*, Strategy 2 is applied in order to add diversity. The details are shown in Table 5. On one hand, the use of Strategy 1 can enable the population to evolve toward the true *PF*; on the other hand, the use of Strategy 2 can effectively maintain the diversity of the population.

Compared with the mutation strategy, Strategy 2 has more advantages. The mutation strategy has almost no prior knowledge. Although it also generates individuals in the less explored areas, the fitness of these individuals may be not sufficiently good for surviving in the selection after the mutation strategy. However, Strategy 2 generates individuals in the less-explored areas and improves them at the same time.

### 3.4 Portability

DRMOS is designed to improve existing MOEAs for the MOPs that can be weakly dimension-reduced in the decision space. DRMOS can be applied as an offspring generation method; that is, DRMOS can be embedded in MOEAs by adding the individuals obtained through DRMOS to the current population, as shown in Figure 2.

In Figure 2, the solid line represents the general flow of MOEAs, and the dotted line represents DRMOS. DRMOS does not affect the flow of MOEAs. Hence, DRMOS can be easily introduced into MOEAs. In this paper, we refer to MOEA XXX with DRMOS as DR_XXX. For example, NSGA-II with DRMOS is called DR_NSGA-II.

UF1 . | UF2 . | UF3 . | UF4 . |
---|---|---|---|

(2.47319,3.33898) | (1.61623,1.34417) | (1.53571,4.64725) | (1.14112,1.15319) |

UF5 | UF6 | UF7 | UF8 |

(6.28565,4.51234) | (2.96617,2.47571) | (3.02704,3.21249) | (2.56648,16.0847,6.02378) |

UF9 | UF10 | ZDT1 | ZDT2 |

(18.8221,16.5887,2.8220) | (16.5010,29.7562,25.3066) | (1.00000,1.17915) | (1.00000,1.00000) |

ZDT3 | ZDT4 | ||

(1.00000,1.00000) | (1.00000,19.8939) |

UF1 . | UF2 . | UF3 . | UF4 . |
---|---|---|---|

(2.47319,3.33898) | (1.61623,1.34417) | (1.53571,4.64725) | (1.14112,1.15319) |

UF5 | UF6 | UF7 | UF8 |

(6.28565,4.51234) | (2.96617,2.47571) | (3.02704,3.21249) | (2.56648,16.0847,6.02378) |

UF9 | UF10 | ZDT1 | ZDT2 |

(18.8221,16.5887,2.8220) | (16.5010,29.7562,25.3066) | (1.00000,1.17915) | (1.00000,1.00000) |

ZDT3 | ZDT4 | ||

(1.00000,1.00000) | (1.00000,19.8939) |

## 4 Simulation Results

In order to evaluate the performance of DRMOS, DRMOS is embedded into several popular MOEAs. The experiment includes three parts: parameter analysis, an experiment on the two memetic local search strategies, and a comparative experiment on benchmark problems.

### 4.1 Metrics

Many metrics can be used for evaluating the performance of MOEAs. Since every metric has its own disadvantages, multiple metrics are employed in our experiments.

#### 4.1.1 Hypervolume

*Hypervolume* (Zitzler and Thiele, 1999) evaluates the size of space in the objective space, covered by non-dominated solutions to a reference point. It can reflect both convergence and maximum spread. In this study, the reference points are set as the maximum values obtained in the results of all the comparative algorithms, as given in Table 6.

#### 4.1.2 Purity

*Purity*(Bandyopadhyay et al., 2004) is used for comparing the convergence ability of comparative algorithms.

*Q*non-dominated solution sets from

*Q*algorithms are included in the comparison, written as .

*R*is the union set of . is the non-dominated solution set of

*R*. is defined as . The purity of the

*i*th algorithm is Equation (10), where

*r*is the number of non-dominated solutions in

_{i}*R*and is the number of non-dominated solutions in . The larger its value, the greater is the convergence that the algorithm has of all the compared algorithms.

_{i}#### 4.1.3 Minimal Spacing

*Minimal Spacing*(Bandyopadhyay et al., 2004) is a modified version of the uniformity metric

*SP*(Van Veldhuizen and Lamont, 2000), as shown in Equation (11). In Equation (11),

*d*is not duplicated, different from the one in

_{i}*SP*. If we take solutions

*j*and

*k*as examples, they are their own nearest neighbors. The distance from

*j*to

*k*is used as both

*d*and

_{j}*d*in

_{k}*SP*. This distance can only be used for one solution in

*Minimal Spacing*. That is, when

*Minimal Spacing*is calculated, all the used distances are marked.

*d*is the nearest distance among the unmarked distances from solution

_{i}*i*.

### 4.2 Parameter Analysis

Compared with other MOEAs, parameter *T* is unique in DRMOS. Therefore, in this section, we analyze the effect of parameter *T* on the behavior of the relation analysis approach. NSGA-II with DRMOS is adopted for the experiment discussed in this section. The original crowding distance has some disadvantages with respect to diversity (Yang et al., 2010). In order to avoid this drawback, the diversity maintenance in Yang et al. (2010) is used. Thus, the algorithm is written as DR_NSGA-II_KN. As the range of *T* is in [0,1], *T* is sampled uniformly by the interval 0.1 for the 2-objective problem UF4 and the 3-objective problem UF8 in 300,000 function evaluations in the experiment. On the one hand, we analyze the behavior of the relation analysis approach by the number of function evaluations and the accuracy rate of the divided subspaces, as shown in Figure 3. On the other hand, we also analyze the influence on the final result using *Hypervolume* and *Minimal Spacing*, as shown in Figure 4.

In Figure 3, the number of function evaluations increases with an increase in *T*. Our relation analysis approach increases the computational cost with an increase in *T*. When *T* is larger than 0.5, the accuracy rate reaches 1. The extra function evaluations are useless. Similarly, the situation is reflected in the final result shown in Figure 4. When *T* is small, the performance of DRMOS is poor because the prediction result has little credibility. Then the subspaces are wrongly divided according to the incorrect information. Finally, the local search turns out to be ineffective. When *T* is larger than 0.5, the performance of DRMOS drops gently because the memetic local search strategy is applied in the correctly divided subspaces. However, the large *T* for sampling incurs a relatively high computational cost. From the above, we can see that there is a trade-off for the selection of *T* between the right mapping relation and the fewest function evaluations.

### 4.3 Experiment on Two Memetic Local Search Strategies

Strategy 1 and Strategy 2 are two memetic local search strategies for MOPs. Especially, Strategy 2 brings a new idea to multi-objective MAs. Therefore, its behavior is analyzed in this section by comparing the DR_NSGA-II_KNs with and without Strategy 2 on UF4 and UF8. All the parameter settings are the same as that in Section 4.2. The results of *Purity*, *Minimal Spacing*, and *Hypervolume* are shown in Table 7. We find that the DR_NSGA-II_KN with Strategy 2 has better performance on both convergence and diversity than the DR_NSGA-II_KN without Strategy 2, especially on the 3-objective problem UF8. Since Strategy 2 can add diversity to the population, it slightly improves the performance of MOEAs.

. | Purity
. | Minimal Spacing
. | Hypervolume
. | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

. | With S2 . | Without S2 . | With S2 . | Without S2 . | With S2 . | Without S2 . | ||||||

Average | Average | Average | Average | Average | Average | |||||||

UF4 | 0.5053 | 0.0224 | 0.4947 | 0.0224 | 0.0095 | 0.0009 | 0.0095 | 0.0007 | 0.3820 | 0.0004 | 0.3818 | 0.0004 |

UF8 | 0.5020 | 0.0182 | 0.4980 | 0.0182 | 0.0686 | 0.0134 | 0.0730 | 0.0122 | 0.8941 | 0.0225 | 0.8918 | 0.0197 |

. | Purity
. | Minimal Spacing
. | Hypervolume
. | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

. | With S2 . | Without S2 . | With S2 . | Without S2 . | With S2 . | Without S2 . | ||||||

Average | Average | Average | Average | Average | Average | |||||||

UF4 | 0.5053 | 0.0224 | 0.4947 | 0.0224 | 0.0095 | 0.0009 | 0.0095 | 0.0007 | 0.3820 | 0.0004 | 0.3818 | 0.0004 |

UF8 | 0.5020 | 0.0182 | 0.4980 | 0.0182 | 0.0686 | 0.0134 | 0.0730 | 0.0122 | 0.8941 | 0.0225 | 0.8918 | 0.0197 |

In order to show the interaction between Strategy 1 (for convergence) and Strategy 2 (for diversity), the average call numbers of Strategy 1 and Strategy 2 of DR_NSGA-II_KN on UF4 and UF8 are recorded in Table 8. For the 2-objective problem UF4, Strategy 1 is called for 75 times, whereas Strategy 2 is called for five times. For the 3-objective problem UF8, Strategy 1 is called for 34 times, whereas Strategy 2 is called for nine times. Strategy 2 is called more times for 3-objective problems than 2-objective problems, because the diversity of 3-objective problems is harder to maintain than that of 2-objective problems. Therefore, the interaction between Strategy 1 and Strategy 2 is robust for different problems.

### 4.4 Benchmark Problems

As DRMOS aims to solve the MOPs that can be weakly dimension-reduced in the decision space, we only employ such problems in the experiment. We choose the problems whose reduction rates are not zero, such as the UF (Zhang et al., 2008b) and ZDT (Zitzler et al., 2000) problems, rather than the DTLZ problems (Deb et al., 2002b). Their reduction rates are shown in Table 9.

UF1UF7 . | UF8UF10 . | ZDT1ZDT4 . | DTLZ1DTLZ4 . |
---|---|---|---|

71.6% | 82.2% | 48.3% | 0% |

UF1UF7 . | UF8UF10 . | ZDT1ZDT4 . | DTLZ1DTLZ4 . |
---|---|---|---|

71.6% | 82.2% | 48.3% | 0% |

In order to present the portability of DRMOS, NSGA-II (Deb et al., 2002a), NSGA-II_KN, SPEA2 (Zitzler et al., 2001), and TDEA (Karahan and Koksalan, 2010) are chosen to embed DRMOS in the comparative experiment. The corresponding algorithms with DRMOS are called DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, and DR_TDEA. In addition, MOEA/D (Zhang et al., 2008b) is well known for its ability to solve complicated MOPs such as the UF problems; hence, it is also included in the comparative algorithms. Finally, the results with respect to the median of the *Hypervolume* in our experiments are presented below. The metrics *Purity*, *Hypervolume*, and *Minimal Spacing* are selected to evaluate results. The experiment parameters are set as shown in Table 10.

Population size n
. | Number of function evaluations . | No. of independent runs . | Crossover probability . | Mutation probability . | Threshold T
. | Territory parameter for TDEA . |
---|---|---|---|---|---|---|

110 (2-objective problems for MOEA/D),g120 (3-objective problems for MOEA/D),100 (other algorithms) | 300,000 including function evaluations of DRMOS | 29 | 1 | 0.1 | 0.5 | 0.01 (2-objective problems),0.1 (3-objective problems) |

Population size n
. | Number of function evaluations . | No. of independent runs . | Crossover probability . | Mutation probability . | Threshold T
. | Territory parameter for TDEA . |
---|---|---|---|---|---|---|

110 (2-objective problems for MOEA/D),g120 (3-objective problems for MOEA/D),100 (other algorithms) | 300,000 including function evaluations of DRMOS | 29 | 1 | 0.1 | 0.5 | 0.01 (2-objective problems),0.1 (3-objective problems) |

#### 4.4.1 Results

##### 4.4.1.1 UF1

UF1 is a complicated problem with 30 decision variables, and its *PS* is a complicated curve restricted by *x*_{1}. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 5. In that figure, DRMOS improves the convergence and diversity of the corresponding MOEAs, among which DRMOS improves TDEA’s diversity the most and SPEA2’s convergence the most. The performance of MOEAs with DRMOS is better than that of MOEA/D. Comparing the results of DR_NSGA-II and DR_NSGA-II_KN, we can see that the uniformity in NSGA-II is improved by the diversity maintenance in Yang et al. (2010).

##### 4.4.1.2 UF2

UF2 is a complicated problem with 30 decision variables, and its *PS* is a complicated curve restricted by *x*_{1}, having a tail with a more complicated curvature than its front. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 6. In that figure, DRMOS slightly improves the diversity of *PF* of these four MOEAs in , where MOEA/D has poor diversity.

##### 4.4.1.3 UF3

UF3 is a complicated problem with 30 decision variables, and its *PS* is a complicated curve restricted by *x*_{1}. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 7. In that figure, DRMOS significantly improves the convergence of all MOEAs, particularly TDEA. DRMOS also improves the diversity of NSGA-II and NSGA-II_KN. For SPEA2, DRMOS slightly improves both its convergence and diversity. MOEA/D maintains good convergence and diversity.

##### 4.4.1.4 UF4

UF4 is a complicated problem with 30 decision variables, and its *PF* is a concave. Further, there are many local optima on its landscape. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 8. In that figure, DRMOS improves MOEAs’ convergence, all the results of MOEAs with DRMOS converge to the true *PF*, and the results of the original MOEAs and MOEA/D all get trapped in local optima. Additionally, the diversity of MOEAs with DRMOS is maintained well, except in the case of DR_SPEA2.

##### 4.4.1.5 UF5

UF5 is a complicated problem with 30 decision variables, and its *PF* consists of 21 discrete points. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, and TDEA are shown in Figure 9. In that figure, DRMOS improves the convergence and diversity of MOEAs so significantly that most of the *PF* are obtained, having a better performance than MOEA/D.

##### 4.4.1.6 UF6

UF6 is a complicated problem with 30 decision variables, and its *PF* is discontinuous. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 10. In that figure, DRMOS cannot improve the convergence of the MOEA significantly. Although it improves the diversity, there is still room for improvement. At the same time, MOEA/D does well in this aspect.

##### 4.4.1.7 UF7

UF7 is a complicated problem with 30 decision variables. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 11. In that figure, the convergence of the original MOEA is already very good, so that DRMOS improves them slightly, but DRMOS does improve their diversity considerably, particularly the diversity of TDEA.

##### 4.4.1.8 UF8

UF8 is a 3-objective problem with 30 decision variables. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 12. In that figure, DRMOS improves the convergence of these four MOEAs so well that their results converge to the true *PF*, which is better than those of MOEA/D. Further, the diversity of MOEAs with DRMOS is improved. However, the diversity of DR_NSGA-II is the worst among the MOEAs with DRMOS, which is caused by the disadvantages of crowding distance.

##### 4.4.1.9 UF9

UF9 is a 3-objective problem with 30 decision variables, and its *PF* is discontinuous. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 13. In that figure, the convergence and diversity of MOEAs are improved by DRMOS, but not satisfactorily. For example, the convergence and diversity of DR_TDEA are a little worse than that of the others.

##### 4.4.1.10 UF10

UF10 is a complicated 3-objective optimization problem with 30 decision variables. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 14. In that figure, DRMOS improves the convergence of four MOEAs so well that their results converge to the true *PF*. Their diversity is improved less by DRMOS. The diversity of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, and DR_TDEA is not satisfactory. Further, neither the convergence nor the diversity of MOEA/D is good in the case of UF10.

##### 4.4.1.11 ZDT1

ZDT1 is a problem with 30 decision variables. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 15. In that figure, all the algorithms have good results of convergence and diversity. The results of DR_NASGA-II and NSGA-II have relatively poor uniformity. Hence, DRMOS cannot improve MOEAs much on ZDT1, and it does not worsen MOEAs on ZDT1, either.

##### 4.4.1.12 ZDT2

ZDT2 is a problem with 30 decision variables, and its *PF* is concave. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 16. In that figure, DRMOS does not make MOEAs perform worse on ZDT2.

##### 4.4.1.13 ZDT3

ZDT3 is a problem with 30 decision variables, and its *PF* is discontinuous. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 17. In that figure, DRMOS does not improve MOEAs on ZDT3.

##### 4.4.1.14 ZDT4

ZDT4 is a problem with 30 decision variables. The resulting *PF*s of DR_NSGA-II, DR_NSGA-II_KN, DR_SPEA2, DR_TDEA, NSGA-II, NSGA-II_KN, SPEA2, TDEA, and MOEA/D are shown in Figure 18. In that figure, DRMOS improves MOEAs slightly on ZDT4.

In order to quantitatively explain the experiment results, the performance measures *Purity*, *Minimal Spacing*, and *Hypervolume* are shown in Figures 19, 20, and 21 in boxplots as a means of statistical analysis, and in Tables 11, 12, and 13, where the winner of the comparison between an MOEA and the MOEA with DRMOS is in bold. As we do not embed DRMOS into MOEA/D, DR_MOEA/D is not applicable in Tables 12 and 13.

. | . | NSGA-II . | NSGA-II_KN . | SPEA2 . | TDEA . | ||||
---|---|---|---|---|---|---|---|---|---|

Average | Average | Average | Average | ||||||

UF1 | DR_XXX | 0.5338 | 0.0367 | 0.5501 | 0.0413 | 0.5448 | 0.0348 | 0.6788 | 0.0560 |

XXX | 0.2719 | 0.0448 | 0.3557 | 0.0367 | 0.3790 | 0.0356 | 0.1697 | 0.0389 | |

MOEA/D | 0.1944 | 0.0510 | 0.0943 | 0.0325 | 0.0762 | 0.0244 | 0.1515 | 0.0446 | |

UF2 | DR_XXX | 0.4715 | 0.0211 | 0.4968 | 0.0198 | 0.4974 | 0.0169 | 0.5911 | 0.0263 |

XXX | 0.2854 | 0.0259 | 0.3608 | 0.0169 | 0.4264 | 0.0178 | 0.2391 | 0.0159 | |

MOEA/D | 0.2430 | 0.0311 | 0.1423 | 0.0220 | 0.0762 | 0.0214 | 0.1698 | 0.0268 | |

UF3 | DR_XXX | 0.6852 | 0.1609 | 0.5780 | 0.1413 | 0.4613 | 0.1629 | 0.4255 | 0.2334 |

XXX | 0.0869 | 0.0723 | 0.2757 | 0.1076 | 0.2683 | 0.0870 | 0.0206 | 0.0161 | |

MOEA/D | 0.2280 | 0.1944 | 0.1464 | 0.1644 | 0.2704 | 0.1747 | 0.5540 | 0.2434 | |

UF4 | DR_XXX | 1.0000 | 0.0000 | 1.0000 | 0.0000 | 1.0000 | 0.0000 | 1.0000 | 0.0000 |

XXX | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

MOEA/D | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

UF5 | DR_XXX | 0.8697 | 0.0891 | 0.8554 | 0.0885 | 0.8409 | 0.0800 | 0.6077 | 0.1834 |

XXX | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

MOEA/D | 0.1303 | 0.0891 | 0.1446 | 0.0885 | 0.1591 | 0.0800 | 0.3923 | 0.1834 | |

UF6 | DR_XXX | 0.0884 | 0.1051 | 0.0719 | 0.0926 | 0.0661 | 0.0981 | 0.0268 | 0.0212 |

XXX | 0.1579 | 0.1646 | 0.1916 | 0.2057 | 0.1756 | 0.1494 | 0.0065 | 0.0083 | |

MOEA/D | 0.7536 | 0.1690 | 0.7365 | 0.1876 | 0.7582 | 0.1855 | 0.9667 | 0.0227 | |

UF7 | DR_XXX | 0.4116 | 0.0200 | 0.4281 | 0.0294 | 0.3596 | 0.0286 | 0.3924 | 0.0536 |

XXX | 0.2734 | 0.0384 | 0.3448 | 0.0520 | 0.4251 | 0.0430 | 0.1866 | 0.0981 | |

MOEA/D | 0.3151 | 0.0351 | 0.2271 | 0.0379 | 0.2153 | 0.0525 | 0.4210 | 0.0549 | |

UF8 | DR_XXX | 0.5115 | 0.0616 | 0.5041 | 0.0549 | 0.5162 | 0.0798 | 0.4887 | 0.0548 |

XXX | 0.1673 | 0.0770 | 0.2219 | 0.0611 | 0.2299 | 0.0905 | 0.2128 | 0.0503 | |

MOEA/D | 0.3213 | 0.0769 | 0.2740 | 0.0632 | 0.2539 | 0.0703 | 0.2985 | 0.0728 | |

UF9 | DR_XXX | 0.6307 | 0.0814 | 0.6833 | 0.1151 | 0.6401 | 0.0832 | 0.3800 | 0.0796 |

XXX | 0.0401 | 0.0296 | 0.0223 | 0.0369 | 0.0268 | 0.0313 | 0.1328 | 0.1142 | |

MOEA/D | 0.3292 | 0.0854 | 0.2943 | 0.1045 | 0.3332 | 0.1007 | 0.4872 | 0.1377 | |

UF10 | DR_XXX | 0.7581 | 0.1440 | 0.7777 | 0.1429 | 0.7828 | 0.1455 | 0.6454 | 0.1952 |

XXX | 0.0025 | 0.0081 | 0.0018 | 0.0068 | 0.0132 | 0.0410 | 0.0005 | 0.0026 | |

MOEA/D | 0.2394 | 0.1428 | 0.2205 | 0.1410 | 0.2040 | 0.1375 | 0.3541 | 0.1955 | |

ZDT1 | DR_XXX | 0.3386 | 0.0039 | 0.3395 | 0.0024 | 0.3384 | 0.0041 | 0.3271 | 0.0061 |

XXX | 0.3390 | 0.0035 | 0.3397 | 0.0024 | 0.3392 | 0.0039 | 0.3316 | 0.0066 | |

MOEA/D | 0.3224 | 0.0073 | 0.3208 | 0.0047 | 0.3224 | 0.0076 | 0.3413 | 0.0076 | |

ZDT2 | DR_XXX | 0.3373 | 0.0019 | 0.3377 | 0.0025 | 0.3372 | 0.0020 | 0.3259 | 0.0065 |

XXX | 0.3374 | 0.0018 | 0.3378 | 0.0023 | 0.3372 | 0.0023 | 0.3309 | 0.0069 | |

MOEA/D | 0.3253 | 0.0036 | 0.3245 | 0.0048 | 0.3256 | 0.0043 | 0.3432 | 0.0045 | |

ZDT3 | DR_XXX | 0.3476 | 0.0120 | 0.3490 | 0.0101 | 0.3449 | 0.0110 | 0.3620 | 0.0214 |

XXX | 0.3544 | 0.0131 | 0.3530 | 0.0109 | 0.3483 | 0.0111 | 0.3814 | 0.0207 | |

MOEA/D | 0.2980 | 0.0220 | 0.2980 | 0.0205 | 0.3067 | 0.0215 | 0.2566 | 0.0216 | |

ZDT4 | DR_XXX | 0.4691 | 0.0201 | 0.4827 | 0.0208 | 0.4834 | 0.0211 | 0.4711 | 0.0280 |

XXX | 0.4703 | 0.0211 | 0.4827 | 0.0208 | 0.4834 | 0.0211 | 0.4769 | 0.0274 | |

MOEA/D | 0.0606 | 0.0412 | 0.0346 | 0.0417 | 0.0333 | 0.0422 | 0.0519 | 0.0538 |

. | . | NSGA-II . | NSGA-II_KN . | SPEA2 . | TDEA . | ||||
---|---|---|---|---|---|---|---|---|---|

Average | Average | Average | Average | ||||||

UF1 | DR_XXX | 0.5338 | 0.0367 | 0.5501 | 0.0413 | 0.5448 | 0.0348 | 0.6788 | 0.0560 |

XXX | 0.2719 | 0.0448 | 0.3557 | 0.0367 | 0.3790 | 0.0356 | 0.1697 | 0.0389 | |

MOEA/D | 0.1944 | 0.0510 | 0.0943 | 0.0325 | 0.0762 | 0.0244 | 0.1515 | 0.0446 | |

UF2 | DR_XXX | 0.4715 | 0.0211 | 0.4968 | 0.0198 | 0.4974 | 0.0169 | 0.5911 | 0.0263 |

XXX | 0.2854 | 0.0259 | 0.3608 | 0.0169 | 0.4264 | 0.0178 | 0.2391 | 0.0159 | |

MOEA/D | 0.2430 | 0.0311 | 0.1423 | 0.0220 | 0.0762 | 0.0214 | 0.1698 | 0.0268 | |

UF3 | DR_XXX | 0.6852 | 0.1609 | 0.5780 | 0.1413 | 0.4613 | 0.1629 | 0.4255 | 0.2334 |

XXX | 0.0869 | 0.0723 | 0.2757 | 0.1076 | 0.2683 | 0.0870 | 0.0206 | 0.0161 | |

MOEA/D | 0.2280 | 0.1944 | 0.1464 | 0.1644 | 0.2704 | 0.1747 | 0.5540 | 0.2434 | |

UF4 | DR_XXX | 1.0000 | 0.0000 | 1.0000 | 0.0000 | 1.0000 | 0.0000 | 1.0000 | 0.0000 |

XXX | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

MOEA/D | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

UF5 | DR_XXX | 0.8697 | 0.0891 | 0.8554 | 0.0885 | 0.8409 | 0.0800 | 0.6077 | 0.1834 |

XXX | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

MOEA/D | 0.1303 | 0.0891 | 0.1446 | 0.0885 | 0.1591 | 0.0800 | 0.3923 | 0.1834 | |

UF6 | DR_XXX | 0.0884 | 0.1051 | 0.0719 | 0.0926 | 0.0661 | 0.0981 | 0.0268 | 0.0212 |

XXX | 0.1579 | 0.1646 | 0.1916 | 0.2057 | 0.1756 | 0.1494 | 0.0065 | 0.0083 | |

MOEA/D | 0.7536 | 0.1690 | 0.7365 | 0.1876 | 0.7582 | 0.1855 | 0.9667 | 0.0227 | |

UF7 | DR_XXX | 0.4116 | 0.0200 | 0.4281 | 0.0294 | 0.3596 | 0.0286 | 0.3924 | 0.0536 |

XXX | 0.2734 | 0.0384 | 0.3448 | 0.0520 | 0.4251 | 0.0430 | 0.1866 | 0.0981 | |

MOEA/D | 0.3151 | 0.0351 | 0.2271 | 0.0379 | 0.2153 | 0.0525 | 0.4210 | 0.0549 | |

UF8 | DR_XXX | 0.5115 | 0.0616 | 0.5041 | 0.0549 | 0.5162 | 0.0798 | 0.4887 | 0.0548 |

XXX | 0.1673 | 0.0770 | 0.2219 | 0.0611 | 0.2299 | 0.0905 | 0.2128 | 0.0503 | |

MOEA/D | 0.3213 | 0.0769 | 0.2740 | 0.0632 | 0.2539 | 0.0703 | 0.2985 | 0.0728 | |

UF9 | DR_XXX | 0.6307 | 0.0814 | 0.6833 | 0.1151 | 0.6401 | 0.0832 | 0.3800 | 0.0796 |

XXX | 0.0401 | 0.0296 | 0.0223 | 0.0369 | 0.0268 | 0.0313 | 0.1328 | 0.1142 | |

MOEA/D | 0.3292 | 0.0854 | 0.2943 | 0.1045 | 0.3332 | 0.1007 | 0.4872 | 0.1377 | |

UF10 | DR_XXX | 0.7581 | 0.1440 | 0.7777 | 0.1429 | 0.7828 | 0.1455 | 0.6454 | 0.1952 |

XXX | 0.0025 | 0.0081 | 0.0018 | 0.0068 | 0.0132 | 0.0410 | 0.0005 | 0.0026 | |

MOEA/D | 0.2394 | 0.1428 | 0.2205 | 0.1410 | 0.2040 | 0.1375 | 0.3541 | 0.1955 | |

ZDT1 | DR_XXX | 0.3386 | 0.0039 | 0.3395 | 0.0024 | 0.3384 | 0.0041 | 0.3271 | 0.0061 |

XXX | 0.3390 | 0.0035 | 0.3397 | 0.0024 | 0.3392 | 0.0039 | 0.3316 | 0.0066 | |

MOEA/D | 0.3224 | 0.0073 | 0.3208 | 0.0047 | 0.3224 | 0.0076 | 0.3413 | 0.0076 | |

ZDT2 | DR_XXX | 0.3373 | 0.0019 | 0.3377 | 0.0025 | 0.3372 | 0.0020 | 0.3259 | 0.0065 |

XXX | 0.3374 | 0.0018 | 0.3378 | 0.0023 | 0.3372 | 0.0023 | 0.3309 | 0.0069 | |

MOEA/D | 0.3253 | 0.0036 | 0.3245 | 0.0048 | 0.3256 | 0.0043 | 0.3432 | 0.0045 | |

ZDT3 | DR_XXX | 0.3476 | 0.0120 | 0.3490 | 0.0101 | 0.3449 | 0.0110 | 0.3620 | 0.0214 |

XXX | 0.3544 | 0.0131 | 0.3530 | 0.0109 | 0.3483 | 0.0111 | 0.3814 | 0.0207 | |

MOEA/D | 0.2980 | 0.0220 | 0.2980 | 0.0205 | 0.3067 | 0.0215 | 0.2566 | 0.0216 | |

ZDT4 | DR_XXX | 0.4691 | 0.0201 | 0.4827 | 0.0208 | 0.4834 | 0.0211 | 0.4711 | 0.0280 |

XXX | 0.4703 | 0.0211 | 0.4827 | 0.0208 | 0.4834 | 0.0211 | 0.4769 | 0.0274 | |

MOEA/D | 0.0606 | 0.0412 | 0.0346 | 0.0417 | 0.0333 | 0.0422 | 0.0519 | 0.0538 |

. | . | NSGA-II . | NSGA-II_KN . | SPEA2 . | TDEA . | MOEA/D . | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Average | Average | Average | Average | Average | |||||||

UF1 | DR_XXX | 0.0095 | 0.0008 | 0.0030 | 0.0002 | 0.0106 | 0.0021 | 0.0101 | 0.0023 | NA | NA |

XXX | 0.0257 | 0.0058 | 0.0327 | 0.0099 | 0.0352 | 0.0094 | 0.0585 | 0.0134 | 0.0039 | 0.0004 | |

UF2 | DR_XXX | 0.0101 | 0.0013 | 0.0030 | 0.0002 | 0.0047 | 0.0008 | 0.0045 | 0.0010 | NA | NA |

XXX | 0.0108 | 0.0011 | 0.0051 | 0.0017 | 0.0089 | 0.0033 | 0.0142 | 0.0047 | 0.0037 | 0.0005 | |

UF3 | DR_XXX | 0.0122 | 0.0023 | 0.0082 | 0.0037 | 0.0228 | 0.0024 | 0.0257 | 0.0049 | NA | NA |

XXX | 0.0265 | 0.0131 | 0.0251 | 0.0063 | 0.0255 | 0.0067 | 0.0364 | 0.0199 | 0.0106 | 0.0043 | |

UF4 | DR_XXX | 0.0114 | 0.0006 | 0.0093 | 0.0007 | 0.0096 | 0.0010 | 0.0070 | 0.0008 | NA | NA |

XXX | 0.0134 | 0.0015 | 0.0092 | 0.0027 | 0.0099 | 0.0020 | 0.0119 | 0.0034 | 0.0094 | 0.0008 | |

UF5 | DR_XXX | 0.0337 | 0.0045 | 0.0322 | 0.0035 | 0.0364 | 0.0053 | 0.0378 | 0.0068 | NA | NA |

XXX | 0.0723 | 0.0305 | 0.0858 | 0.0385 | 0.0917 | 0.0419 | 0.1557 | 0.1276 | 0.0331 | 0.0037 | |

UF6 | DR_XXX | 0.0520 | 0.0082 | 0.0535 | 0.0074 | 0.0565 | 0.0080 | 0.0977 | 0.0142 | NA | NA |

XXX | 0.0480 | 0.0416 | 0.0409 | 0.0374 | 0.0545 | 0.0492 | 0.1064 | 0.0927 | 0.0536 | 0.0067 | |

UF7 | DR_XXX | 0.0100 | 0.0006 | 0.0028 | 0.0002 | 0.0155 | 0.0040 | 0.0168 | 0.0036 | NA | NA |

XXX | 0.0220 | 0.0028 | 0.0224 | 0.0051 | 0.0252 | 0.0042 | 0.0335 | 0.0117 | 0.0058 | 0.0007 | |

UF8 | DR_XXX | 0.1047 | 0.0133 | 0.0709 | 0.0135 | 0.0716 | 0.0113 | 0.0764 | 0.0122 | NA | NA |

XXX | 0.2325 | 0.0734 | 0.1734 | 0.0681 | 0.1603 | 0.0519 | 0.1731 | 0.0739 | 0.0750 | 0.012 | |

UF9 | DR_XXX | 0.0906 | 0.0240 | 0.0818 | 0.0216 | 0.0907 | 0.0220 | 0.1143 | 0.0325 | NA | NA |

XXX | 0.4581 | 0.1874 | 0.4522 | 0.2242 | 0.5046 | 0.2216 | 0.2163 | 0.0692 | 0.1009 | 0.0200 | |

UF10 | DR_XXX | 0.1230 | 0.0208 | 0.1095 | 0.0156 | 0.1165 | 0.0223 | 0.1322 | 0.0288 | NA | NA |

XXX | 0.2417 | 0.1069 | 0.2302 | 0.1612 | 0.1585 | 0.0541 | 0.2230 | 0.0731 | 0.1110 | 0.0193 | |

ZDT1 | DR_XXX | 0.0110 | 0.0012 | 0.0033 | 0.0001 | 0.0038 | 0.0003 | 0.0042 | 0.0027 | NA | NA |

XXX | 0.0102 | 0.0010 | 0.0029 | 0.0002 | 0.0034 | 0.0002 | 0.0036 | 0.0002 | 0.0039 | 0.0005 | |

ZDT2 | DR_XXX | 0.0112 | 0.0015 | 0.0033 | 0.0001 | 0.0038 | 0.0004 | 0.0037 | 0.0003 | NA | NA |

XXX | 0.0103 | 0.0008 | 0.0030 | 0.0002 | 0.0034 | 0.0002 | 0.0037 | 0.0003 | 0.0039 | 0.0006 | |

ZDT3 | DR_XXX | 0.0290 | 0.0007 | 0.0262 | 0.0000 | 0.0277 | 0.0032 | 0.0257 | 0.0028 | NA | NA |

XXX | 0.0287 | 0.0021 | 0.0260 | 0.0008 | 0.0268 | 0.0013 | 0.0245 | 0.0022 | 0.0280 | 0.0018 | |

ZDT4 | DR_XXX | 0.0108 | 0.0011 | 0.0033 | 0.0001 | 0.0072 | 0.0101 | 0.0039 | 0.0003 | NA | NA |

XXX | 0.0100 | 0.0008 | 0.0029 | 0.0004 | 0.0035 | 0.0004 | 0.0696 | 0.3607 | 0.0041 | 0.0007 |

. | . | NSGA-II . | NSGA-II_KN . | SPEA2 . | TDEA . | MOEA/D . | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Average | Average | Average | Average | Average | |||||||

UF1 | DR_XXX | 0.0095 | 0.0008 | 0.0030 | 0.0002 | 0.0106 | 0.0021 | 0.0101 | 0.0023 | NA | NA |

XXX | 0.0257 | 0.0058 | 0.0327 | 0.0099 | 0.0352 | 0.0094 | 0.0585 | 0.0134 | 0.0039 | 0.0004 | |

UF2 | DR_XXX | 0.0101 | 0.0013 | 0.0030 | 0.0002 | 0.0047 | 0.0008 | 0.0045 | 0.0010 | NA | NA |

XXX | 0.0108 | 0.0011 | 0.0051 | 0.0017 | 0.0089 | 0.0033 | 0.0142 | 0.0047 | 0.0037 | 0.0005 | |

UF3 | DR_XXX | 0.0122 | 0.0023 | 0.0082 | 0.0037 | 0.0228 | 0.0024 | 0.0257 | 0.0049 | NA | NA |

XXX | 0.0265 | 0.0131 | 0.0251 | 0.0063 | 0.0255 | 0.0067 | 0.0364 | 0.0199 | 0.0106 | 0.0043 | |

UF4 | DR_XXX | 0.0114 | 0.0006 | 0.0093 | 0.0007 | 0.0096 | 0.0010 | 0.0070 | 0.0008 | NA | NA |

XXX | 0.0134 | 0.0015 | 0.0092 | 0.0027 | 0.0099 | 0.0020 | 0.0119 | 0.0034 | 0.0094 | 0.0008 | |

UF5 | DR_XXX | 0.0337 | 0.0045 | 0.0322 | 0.0035 | 0.0364 | 0.0053 | 0.0378 | 0.0068 | NA | NA |

XXX | 0.0723 | 0.0305 | 0.0858 | 0.0385 | 0.0917 | 0.0419 | 0.1557 | 0.1276 | 0.0331 | 0.0037 | |

UF6 | DR_XXX | 0.0520 | 0.0082 | 0.0535 | 0.0074 | 0.0565 | 0.0080 | 0.0977 | 0.0142 | NA | NA |

XXX | 0.0480 | 0.0416 | 0.0409 | 0.0374 | 0.0545 | 0.0492 | 0.1064 | 0.0927 | 0.0536 | 0.0067 | |

UF7 | DR_XXX | 0.0100 | 0.0006 | 0.0028 | 0.0002 | 0.0155 | 0.0040 | 0.0168 | 0.0036 | NA | NA |

XXX | 0.0220 | 0.0028 | 0.0224 | 0.0051 | 0.0252 | 0.0042 | 0.0335 | 0.0117 | 0.0058 | 0.0007 | |

UF8 | DR_XXX | 0.1047 | 0.0133 | 0.0709 | 0.0135 | 0.0716 | 0.0113 | 0.0764 | 0.0122 | NA | NA |

XXX | 0.2325 | 0.0734 | 0.1734 | 0.0681 | 0.1603 | 0.0519 | 0.1731 | 0.0739 | 0.0750 | 0.012 | |

UF9 | DR_XXX | 0.0906 | 0.0240 | 0.0818 | 0.0216 | 0.0907 | 0.0220 | 0.1143 | 0.0325 | NA | NA |

XXX | 0.4581 | 0.1874 | 0.4522 | 0.2242 | 0.5046 | 0.2216 | 0.2163 | 0.0692 | 0.1009 | 0.0200 | |

UF10 | DR_XXX | 0.1230 | 0.0208 | 0.1095 | 0.0156 | 0.1165 | 0.0223 | 0.1322 | 0.0288 | NA | NA |

XXX | 0.2417 | 0.1069 | 0.2302 | 0.1612 | 0.1585 | 0.0541 | 0.2230 | 0.0731 | 0.1110 | 0.0193 | |

ZDT1 | DR_XXX | 0.0110 | 0.0012 | 0.0033 | 0.0001 | 0.0038 | 0.0003 | 0.0042 | 0.0027 | NA | NA |

XXX | 0.0102 | 0.0010 | 0.0029 | 0.0002 | 0.0034 | 0.0002 | 0.0036 | 0.0002 | 0.0039 | 0.0005 | |

ZDT2 | DR_XXX | 0.0112 | 0.0015 | 0.0033 | 0.0001 | 0.0038 | 0.0004 | 0.0037 | 0.0003 | NA | NA |

XXX | 0.0103 | 0.0008 | 0.0030 | 0.0002 | 0.0034 | 0.0002 | 0.0037 | 0.0003 | 0.0039 | 0.0006 | |

ZDT3 | DR_XXX | 0.0290 | 0.0007 | 0.0262 | 0.0000 | 0.0277 | 0.0032 | 0.0257 | 0.0028 | NA | NA |

XXX | 0.0287 | 0.0021 | 0.0260 | 0.0008 | 0.0268 | 0.0013 | 0.0245 | 0.0022 | 0.0280 | 0.0018 | |

ZDT4 | DR_XXX | 0.0108 | 0.0011 | 0.0033 | 0.0001 | 0.0072 | 0.0101 | 0.0039 | 0.0003 | NA | NA |

XXX | 0.0100 | 0.0008 | 0.0029 | 0.0004 | 0.0035 | 0.0004 | 0.0696 | 0.3607 | 0.0041 | 0.0007 |

. | . | NSGA-II . | NSGA-II_KN . | SPEA2 . | TDEA . | MOEA/D . | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Average | Average | Average | Average | Average | |||||||

UF1 | DR_XXX | 7.9131e+0 | 1.2946e-3 | 7.9167e+0 | 1.1900e-3 | 7.9163e+0 | 1.5431e-3 | 7.9159e+0 | 3.0492e-3 | NA | NA |

XXX | 7.8543e+0 | 8.3469e-2 | 7.8478e+0 | 9.1612e-2 | 7.8139e+0 | 8.5622e-2 | 7.7337e+0 | 1.6247e-1 | 7.8803e+0 | 1.6993e-2 | |

UF2 | DR_XXX | 1.8284e+0 | 6.4263e-4 | 1.8322e+0 | 5.9349e-4 | 1.8338e+0 | 3.1677e-4 | 1.8336e+0 | 2.7505e-4 | NA | NA |

XXX | 1.8234e+0 | 2.1762e-3 | 1.8281e+0 | 1.5463e-3 | 1.8270e+0 | 2.9476e-3 | 1.7798e+0 | 8.1234e-3 | 1.8025e+0 | 8.2006e-3 | |

UF3 | DR_XXX | 6.7854e+0 | 4.5086e-3 | 6.7886e+0 | 3.2226e-3 | 6.7731e+0 | 2.0519e-3 | 6.7695e+0 | 3.5410e-3 | NA | NA |

XXX | 6.5128e+0 | 5.2076e-1 | 6.5884e+0 | 6.5639e-2 | 6.5886e+0 | 6.4911e-2 | 6.0922e+0 | 8.4377e-1 | 6.6602e+0 | 2.4319e-1 | |

UF4 | DR_XXX | 6.2901e-1 | 4.8576e-4 | 6.3014e-1 | 7.0708e-4 | 6.2988e-1 | 6.7927e-4 | 6.2982e-1 | 6.5975e-4 | NA | NA |

XXX | 5.3176e-1 | 8.3824e-3 | 5.3696e-1 | 8.3331e-3 | 5.3871e-1 | 7.8182e-3 | 5.2743e-1 | 8.4661e-3 | 4.9125e-1 | 1.5925e-2 | |

UF5 | DR_XXX | 2.7699e+1 | 1.0814e-1 | 2.7716e+1 | 8.9045e-2 | 2.7648e+1 | 1.5418e-1 | 2.7665e+1 | 1.2687e-1 | NA | NA |

XXX | 2.2902e+1 | 1.1229e+0 | 2.2729e+1 | 1.2970e+0 | 2.2406e+1 | 1.1283e+0 | 2.1620e+1 | 1.0950e+0 | 2.6391e+1 | 1.2971e+0 | |

UF6 | DR_XXX | 6.5450e+0 | 7.3190e-2 | 6.5442e+0 | 7.3465e-2 | 6.5529e+0 | 6.5465e-2 | 6.5429e+0 | 6.5820e-2 | NA | NA |

XXX | 5.4665e+0 | 7.8403e-1 | 5.2870e+0 | 6.2662e-1 | 5.4305e+0 | 6.8668e-1 | 5.1661e+0 | 5.6455e-1 | 6.7192e+0 | 1.2797e-1 | |

UF7 | DR_XXX | 9.2127e+0 | 5.2528e-4 | 9.2166e+0 | 6.9432e-4 | 9.2061e+0 | 3.7191e-3 | 9.2052e+0 | 4.5628e-3 | NA | NA |

XXX | 9.1915e+0 | 2.5094e-2 | 9.1933e+0 | 2.7202e-2 | 9.0740e+0 | 4.6109e-1 | 8.7229e+0 | 8.9918e-1 | 9.0086e+0 | 6.0469e-1 | |

UF8 | DR_XXX | 2.4781e+2 | 1.8651e-1 | 2.4718e+2 | 2.8136e+0 | 2.4686e+2 | 4.6795e+0 | 2.4783e+2 | 2.5708e-1 | NA | NA |

XXX | 2.4445e+2 | 4.9390e+0 | 2.4780e+2 | 8.2368e-2 | 2.4623e+2 | 3.8505e+0 | 2.4075e+2 | 5.0347e+0 | 2.3686e+2 | 7.1091e+0 | |

UF9 | DR_XXX | 8.8032e+2 | 4.6104e-1 | 8.8059e+2 | 3.9150e-1 | 8.8077e+2 | 8.4689e-2 | 8.8028e+2 | 8.9085e-1 | NA | NA |

XXX | 8.6613e+2 | 5.8196e+0 | 8.7017e+2 | 4.8483e+0 | 8.6879e+2 | 3.8653e+0 | 8.6705e+2 | 3.5323e+0 | 8.6965e+2 | 3.4293e+0 | |

UF10 | DR_XXX | 1.2419e+4 | 2.3643e+1 | 1.2404e+4 | 8.3769e+1 | 1.2419e+4 | 2.1024e+1 | 1.2422e+4 | 1.8994e+0 | NA | NA |

XXX | 1.1700e+4 | 2.0852e+2 | 1.1686e+4 | 1.8232e+2 | 1.1697e+4 | 2.1475e+2 | 1.1598e+4 | 1.1099e+2 | 1.1647e+4 | 1.3973e+2 | |

ZDT1 | DR_XXX | 8.3872e-1 | 5.2571e-4 | 8.4095e-1 | 3.9413e-5 | 8.4085e-1 | 5.4778e-5 | 8.4041e-1 | 1.3518e-4 | NA | NA |

XXX | 8.3904e-1 | 4.8790e-4 | 8.4100e-1 | 3.4338e-5 | 8.4095e-1 | 5.5007e-5 | 8.4050e-1 | 1.5482e-4 | 8.3945e-1 | 1.0314e-4 | |

ZDT2 | DR_XXX | 3.2637e-1 | 6.0622e-4 | 3.2851e-1 | 3.3327e-5 | 3.2847e-1 | 9.2658e-5 | 3.2803e-1 | 1.4160e-4 | NA | NA |

XXX | 3.2664e-1 | 3.2795e-4 | 3.2856e-1 | 3.7613e-5 | 3.2851e-1 | 4.9436e-5 | 3.2809e-1 | 1.2634e-4 | 3.2801e-1 | 3.3747e-5 | |

ZDT3 | DR_XXX | 8.8754e-1 | 2.6749e-4 | 8.8842e-1 | 5.0012e-5 | 8.8673e-1 | 1.4302e-3 | 8.8544e-1 | 1.1316e-2 | NA | NA |

XXX | 8.8550e-1 | 1.1279e-2 | 8.8823e-1 | 1.1977e-3 | 8.8646e-1 | 1.8861e-3 | 8.8731e-1 | 2.3451e-3 | 8.8381e-1 | 2.1779e-4 | |

ZDT4 | DR_XXX | 1.9553e+1 | 5.4229e-4 | 1.9556e+1 | 2.7921e-5 | 1.9552e+1 | 1.2566e-2 | 1.9555e+1 | 1.3772e-4 | NA | NA |

XXX | 1.9554e+1 | 3.3280e-4 | 1.9549e+1 | 2.7062e-2 | 1.9556e+1 | 3.8367e-4 | 1.9555e+1 | 1.3688e-4 | 1.9544e+1 | 3.9139e-3 |

. | . | NSGA-II . | NSGA-II_KN . | SPEA2 . | TDEA . | MOEA/D . | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Average | Average | Average | Average | Average | |||||||

UF1 | DR_XXX | 7.9131e+0 | 1.2946e-3 | 7.9167e+0 | 1.1900e-3 | 7.9163e+0 | 1.5431e-3 | 7.9159e+0 | 3.0492e-3 | NA | NA |

XXX | 7.8543e+0 | 8.3469e-2 | 7.8478e+0 | 9.1612e-2 | 7.8139e+0 | 8.5622e-2 | 7.7337e+0 | 1.6247e-1 | 7.8803e+0 | 1.6993e-2 | |

UF2 | DR_XXX | 1.8284e+0 | 6.4263e-4 | 1.8322e+0 | 5.9349e-4 | 1.8338e+0 | 3.1677e-4 | 1.8336e+0 | 2.7505e-4 | NA | NA |

XXX | 1.8234e+0 | 2.1762e-3 | 1.8281e+0 | 1.5463e-3 | 1.8270e+0 | 2.9476e-3 | 1.7798e+0 | 8.1234e-3 | 1.8025e+0 | 8.2006e-3 | |

UF3 | DR_XXX | 6.7854e+0 | 4.5086e-3 | 6.7886e+0 | 3.2226e-3 | 6.7731e+0 | 2.0519e-3 | 6.7695e+0 | 3.5410e-3 | NA | NA |

XXX | 6.5128e+0 | 5.2076e-1 | 6.5884e+0 | 6.5639e-2 | 6.5886e+0 | 6.4911e-2 | 6.0922e+0 | 8.4377e-1 | 6.6602e+0 | 2.4319e-1 | |

UF4 | DR_XXX | 6.2901e-1 | 4.8576e-4 | 6.3014e-1 | 7.0708e-4 | 6.2988e-1 | 6.7927e-4 | 6.2982e-1 | 6.5975e-4 | NA | NA |

XXX | 5.3176e-1 | 8.3824e-3 | 5.3696e-1 | 8.3331e-3 | 5.3871e-1 | 7.8182e-3 | 5.2743e-1 | 8.4661e-3 | 4.9125e-1 | 1.5925e-2 | |

UF5 | DR_XXX | 2.7699e+1 | 1.0814e-1 | 2.7716e+1 | 8.9045e-2 | 2.7648e+1 | 1.5418e-1 | 2.7665e+1 | 1.2687e-1 | NA | NA |

XXX | 2.2902e+1 | 1.1229e+0 | 2.2729e+1 | 1.2970e+0 | 2.2406e+1 | 1.1283e+0 | 2.1620e+1 | 1.0950e+0 | 2.6391e+1 | 1.2971e+0 | |

UF6 | DR_XXX | 6.5450e+0 | 7.3190e-2 | 6.5442e+0 | 7.3465e-2 | 6.5529e+0 | 6.5465e-2 | 6.5429e+0 | 6.5820e-2 | NA | NA |

XXX | 5.4665e+0 | 7.8403e-1 | 5.2870e+0 | 6.2662e-1 | 5.4305e+0 | 6.8668e-1 | 5.1661e+0 | 5.6455e-1 | 6.7192e+0 | 1.2797e-1 | |

UF7 | DR_XXX | 9.2127e+0 | 5.2528e-4 | 9.2166e+0 | 6.9432e-4 | 9.2061e+0 | 3.7191e-3 | 9.2052e+0 | 4.5628e-3 | NA | NA |

XXX | 9.1915e+0 | 2.5094e-2 | 9.1933e+0 | 2.7202e-2 | 9.0740e+0 | 4.6109e-1 | 8.7229e+0 | 8.9918e-1 | 9.0086e+0 | 6.0469e-1 | |

UF8 | DR_XXX | 2.4781e+2 | 1.8651e-1 | 2.4718e+2 | 2.8136e+0 | 2.4686e+2 | 4.6795e+0 | 2.4783e+2 | 2.5708e-1 | NA | NA |

XXX | 2.4445e+2 | 4.9390e+0 | 2.4780e+2 | 8.2368e-2 | 2.4623e+2 | 3.8505e+0 | 2.4075e+2 | 5.0347e+0 | 2.3686e+2 | 7.1091e+0 | |

UF9 | DR_XXX | 8.8032e+2 | 4.6104e-1 | 8.8059e+2 | 3.9150e-1 | 8.8077e+2 | 8.4689e-2 | 8.8028e+2 | 8.9085e-1 | NA | NA |

XXX | 8.6613e+2 | 5.8196e+0 | 8.7017e+2 | 4.8483e+0 | 8.6879e+2 | 3.8653e+0 | 8.6705e+2 | 3.5323e+0 | 8.6965e+2 | 3.4293e+0 | |

UF10 | DR_XXX | 1.2419e+4 | 2.3643e+1 | 1.2404e+4 | 8.3769e+1 | 1.2419e+4 | 2.1024e+1 | 1.2422e+4 | 1.8994e+0 | NA | NA |

XXX | 1.1700e+4 | 2.0852e+2 | 1.1686e+4 | 1.8232e+2 | 1.1697e+4 | 2.1475e+2 | 1.1598e+4 | 1.1099e+2 | 1.1647e+4 | 1.3973e+2 | |

ZDT1 | DR_XXX | 8.3872e-1 | 5.2571e-4 | 8.4095e-1 | 3.9413e-5 | 8.4085e-1 | 5.4778e-5 | 8.4041e-1 | 1.3518e-4 | NA | NA |

XXX | 8.3904e-1 | 4.8790e-4 | 8.4100e-1 | 3.4338e-5 | 8.4095e-1 | 5.5007e-5 | 8.4050e-1 | 1.5482e-4 | 8.3945e-1 | 1.0314e-4 | |

ZDT2 | DR_XXX | 3.2637e-1 | 6.0622e-4 | 3.2851e-1 | 3.3327e-5 | 3.2847e-1 | 9.2658e-5 | 3.2803e-1 | 1.4160e-4 | NA | NA |

XXX | 3.2664e-1 | 3.2795e-4 | 3.2856e-1 | 3.7613e-5 | 3.2851e-1 | 4.9436e-5 | 3.2809e-1 | 1.2634e-4 | 3.2801e-1 | 3.3747e-5 | |

ZDT3 | DR_XXX | 8.8754e-1 | 2.6749e-4 | 8.8842e-1 | 5.0012e-5 | 8.8673e-1 | 1.4302e-3 | 8.8544e-1 | 1.1316e-2 | NA | NA |

XXX | 8.8550e-1 | 1.1279e-2 | 8.8823e-1 | 1.1977e-3 | 8.8646e-1 | 1.8861e-3 | 8.8731e-1 | 2.3451e-3 | 8.8381e-1 | 2.1779e-4 | |

ZDT4 | DR_XXX | 1.9553e+1 | 5.4229e-4 | 1.9556e+1 | 2.7921e-5 | 1.9552e+1 | 1.2566e-2 | 1.9555e+1 | 1.3772e-4 | NA | NA |

XXX | 1.9554e+1 | 3.3280e-4 | 1.9549e+1 | 2.7062e-2 | 1.9556e+1 | 3.8367e-4 | 1.9555e+1 | 1.3688e-4 | 1.9544e+1 | 3.9139e-3 |

*Purity* represents convergence. The aim of our comparative experiments is to analyze the improvement of DRMOS embedded in classic MOEAs. The comparisons are performed between MOEAs and the MOEAs with DRMOS. Thus, *Purity* is calculated from the results of three algorithms: an MOEA, the MOEA with DRMOS, and MOEA/D. The results are shown in four groups in Table 11. From the boxplots in Figure 19, it is clear that DRMOS increases the convergence abilities of MOEAs on the UF problems. The *Purity* of MOEAs with DRMOS has a better value than their corresponding MOEAs and MOEA/D, particularly MOEAs with DRMOS outperforming the corresponding MOEAs on UF4 and UF5. For the ZDT problems, DRMOS has no improvement on convergence. Sometimes, the *Purity* of MOEAs with DRMOS is a little worse, but not significantly.

*Minimal Spacing* is employed to evaluate the uniformity of results. From the boxplots shown in Figure 20 and the data given in Table 12, DRMOS improves the uniformity of the corresponding MOEAs for the UF problems, although different MOEAs have a different uniformity ability. The uniformity of MOEAs decreased slightly for the ZDT problems but this does not affect their *PF*s significantly.

*Hypervolume* is a metric for evaluating both convergence and maximum spread, and can be used as a general performance measure. From the boxplots shown in Figure 21 and data given in Table 13, DRMOS improves MOEAs in the case of the UF problems because the *Hypervolume* of MOEAs with DRMOS is larger than that of the corresponding MOEAs. However, as the *Hypervolume* of the MOEAs with DRMOS and the corresponding MOEAs are similar, the performance on the ZDT problems cannot be improved by DRMOS.

#### 4.4.2 Discussion

In the case of the UF problems, DRMOS reduces the dimension of the decision space by sampling, and it adopts two memetic local search strategies in the divided decision subspaces. DRMOS increases the convergence speed and diversity, its performance is even better than that of MOEA/D. DRMOS cannot improve the performance of MOEAs on UF6 because of the unsuitable local search strategies. For the relatively simpler ZDT problems, the advantages of DRMOS are not obvious. Because the reduction rates of the ZDT problems are small, the dimension cannot be reduced significantly. DRMOS improves MOEAs slightly in the case of the ZDT problems. Furthermore, the relation analysis approach wastes some function evaluations in the meantime, which leads to a slightly worse performance than that of the original MOEAs.

In general, DRMOS significantly improves convergence, diversity, and uniformity for different MOEAs. Although different MOEAs have different advantages and disadvantages, DRMOS improves the search ability on the MOPs that can be weakly dimension-reduced in the decision space. Moreover, DRMOA can be easily embedded in existing MOEAs to improve their performance. For the MOPs that cannot be weakly dimension-reduced in the decision space, or the ones with small dimension reduction rates, the MOEAs with DRMOS degenerate to their original algorithms.

## 5 Conclusion

A memetic optimization strategy based on dimension reduction in decision space is proposed in this paper. This strategy reduces the dimension of the decision space by a relation analysis approach and two memetic local search strategies in order to improve performance. DRMOS has good portability to existing MOEAs. The work discussed in this paper has three major contributions as follows.

**Relation Analysis.**DRMOS applies a simple model of sampling to obtain the mapping relation between decision variables and objective functions, which provides the knowledge for dimension reduction in the decision space.**Memetic Local Search Strategy.**Memetic local search strategies in DRMOS are employed in the divided decision subspace, which decompose the complicated problem with high dimensions into several simple problems with low dimensions. Two strategies aim at convergence and diversity, respectively.**Portability.**DRMOS can be embedded easily in existing MOEAs in order to apply its powerful performance to the MOPs that can be weakly dimension-reduced in the decision space, which has no influence on the other types of problems for the original performance of MOEAs.

DRMOS takes advantage of the information in the process of optimization to guide the search. Although its superior performance has been shown by experiments, DRMOS still has some disadvantages. (1) DRMOS is only effective in the case of the MOPs that can be weakly dimension-reduced. It degenerates to the original algorithm on the MOPs that cannot be weakly dimension-reduced in the decision space. As for the problems that cannot be divided into subspaces, the memetic local search strategy is invalid. (2) Its performance is not satisfactory for the problems with discontinuous *PF*s, because the computational cost is not self-adaptively assigned according to the different situations of *PF*s in DRMOS. The self-adaptive computational cost assignment is another topic of our future research. (3) The results show that DRMOS is not effective for the problems with small reduction rates, because the relation analysis approach costs many function evaluations rather than searching. Furthermore, the current relation analysis approach is based on a relatively simple model. Some advanced technology such as statistical learning methods and relation mining should be used in DRMOS.

## Acknowledgments

This work was partially supported by the National Basic Research Program (973 Program) of China, under Grant 2013CB329402, an EU FP7 IRSES, under Grant 247619, the National Natural Science Foundation of China, under Grants 61371201 and 61272279, the National Research Foundation for the Doctoral Program of Higher Education of China, under Grant 20100203120008, the Fund for Foreign Scholars in University Research and Teaching Programs, under Grant B07048, and the Program for Cheung Kong Scholars and Innovative Research Team in University under Grant IRT1170. The authors are grateful for Xin Yao’s comments on the paper.

## References

*n*-dimensional landscapes

*k*-nearest neighbor list-based immune multi-objective optimization