# Objective reduction based on nonlinear correlation information entropy

- First Online:

## Abstract

It is hard to obtain the entire solution set of a many-objective optimization problem (MaOP) by multi-objective evolutionary algorithms (MOEAs) because of the difficulties brought by the large number of objectives. However, the redundancy of objectives exists in some problems with correlated objectives (linearly or nonlinearly). Objective reduction can be used to decrease the difficulties of some MaOPs. In this paper, we propose a novel objective reduction approach based on nonlinear correlation information entropy (NCIE). It uses the NCIE matrix to measure the linear and nonlinear correlation between objectives and a simple method to select the most conflicting objectives during the execution of MOEAs. We embed our approach into both Pareto-based and indicator-based MOEAs to analyze the impact of our reduction method on the performance of these algorithms. The results show that our approach significantly improves the performance of Pareto-based MOEAs on both reducible and irreducible MaOPs, but does not much help the performance of indicator-based MOEAs.

### Keywords

Multi-objective optimization Objective reduction Nonlinear correlation information entropy Multi-objective evolutionary algorithm Dimension reduction## 1 Introduction

Loosely speaking, a many-objective optimization problem (MaOP) (Khare et al. 2003; Praditwong and Yao 2007) is a special kind of multi-objective optimization problem (MOP) with more than three objectives (Fleming et al. 2005; Hughes 2007). The large number of objectives in many-objective optimization brings many challenges to multi-objective evolutionary algorithms (MOEAs). Most solutions in the population of an MaOP are non-dominated (Ishibuchi et al. 2008), thus, the selection mechanism based on the Pareto dominance is less effective. Pareto-based MOEAs such as NSGA-II (Deb et al. 2002a) fail to solve MaOPs (Purshouse and Fleming 2003; Hughes 2005; Wagner et al. 2007; Khare et al. 2003). Without an effective dominance relation, MOEAs are unable to provide promising search directions (Deb and Jain 2014). Moreover, the growing number of objectives increases the computational complexity of Pareto-based MOEAs. Although the non-dominated rank sort (Deb and Tiwari 2005), deductive sort (McClymont and Keedwell 2012), and corner sort (Wang and Yao 2014) have been proposed to reduce that complexity, the progress is still unsatisfactory.

*Dominance relation modification*The Pareto dominance is ineffective for MaOPs. Much work aims to modify the original dominance relation (Köppen et al. 2005; Kukkonen and Lampinen 2007; Sato et al. 2007; Farina and Amato 2002; Dai et al. 2015), but their performance is still less than satisfactory.*Decomposition-based MOEAs*The main idea is to solve MaOPs by aggregation functions with a series of weight vectors to obtain several single-objective optimization problems (Zhang and Li 2007; Ma et al. 2014), but it suffers from poor performance on MaOPs with highly correlated objectives (Ishibuchi et al. 2009) because of the unsuitable arrangement of weight vectors (Ishibuchi et al. 2011a, b).*Indicator-based MOEAs*With a metric as a single objective, indicator-based MOEAs can avoid employing the Pareto dominance (Zitzler and Künzli 2004; Bader and Zitzler 2011; Wagner et al. 2007; Gong et al. 2014). However, \(I_{\varepsilon +}\) in IBEA Zitzler and Künzli (2004) and \(I_{H}\) in HypE Bader and Zitzler (2011) provide unsatisfactory diversity on the Pareto front (PF) (Hadka and Reed 2012).*Incorporation with decision makers*Usually, decision makers do not need all the optimal solutions of MaOPs (Cvetkovic and Parmee 2002). They can input their interested regions or preferences to obtain parts of the non-dominated solution set (Sindhya et al. 2011; Ben Said et al. 2010; Koksalan and Karahan 2010; Wang et al. 2013a; Kim et al. 2012; Karahan and Koksalan 2010; Giagkiozis and Fleming 2014). Additionally, decision makers have different targets for different objectives and multi-target search was employed (Wang et al. 2013b).*Objective reduction*For some MOPs, unnecessary objectives can be ignored without changing their Pareto sets (Gal and Hanne 1999). Thus, the difficulty caused by a large number of objectives of a MaOP can be reduced (Fonseca and Fleming 1995; Coello Coello 2005; Deb 2001), and the existing Pareto-based MOEAs for MOPs with low-dimensional objectives can be used.

*Dominance relation preservation-based objective reduction*It is based on a measure for the changes of the dominance structure, which obtains a minimum subset of objectives with the preserved dominance relation (Brockhoff and Zitzler 2006). An additional term \(\delta \) is adopted to measure the difference between the dominance structures of two subsets. However, the technique can only be applied to the linear objective reduction.*Pareto corner search*The Pareto corner search evolutionary algorithm (PCSEA) (Singh et al. 2011) is a newly proposed objective reduction approach. It only searches the corners of PFs. Then, it uses the obtained solutions to analyze the relation among objectives. Finally, it outputs a subset of non-correlated objectives. PCSEA is an off-line method.*Machine learning-based objective reduction*As the process of objective reduction can be seen as feature selection, this method focuses on the objectives with negative correlation and uses an improved correlation matrix of objectives to measure the conflict degree of two objectives (López Jaimes et al. 2008). With the obtained correlation matrix as distances, the method divides those objectives into neighborhoods. Then, it adopts a \(q\)-neighbor structure to select objectives. However, \(q\) has to be set in advance. Other machine learning techniques for dimension reduction, such as principal component analysis (PCA) and maximum variance unfolding (MVU), have also been applied to objective reduction (Saxena and Deb 2007; Deb and Saxena 2005). These objective reduction methods use machine learning techniques to select conflicting objectives according to the correlation information (the correlation matrix and correntropy matrix, for instance).

Objectives that can be reduced are either linearly or nonlinearly correlated, mostly nonlinearly correlated (Saxena et al. 2013). However, the majority of the existing approaches use linear statistical tools to measure both linear and nonlinear correlation. In such cases, nonlinear correlation would be weakened by the linear description, which misleads the reduction.

In this paper, we use the same measurement for both linear and nonlinear correlation; thus, the performance of online objective reduction approaches can be improved. We find NCIE (Wang et al. 2005) to be a very robust measure for both linearly and nonlinearly correlated datasets, which has been applied to the analysis of neurophysiological signals (Pereda et al. 2005), the quantification of the dependence among noisy data (Khan et al. 2007), etc. Therefore, we adopt NCIE as a correlation measure in objective reduction and study its impact on online objective reduction approaches.

The rest of the paper is organized as follows. We first show different cases of redundant objectives in MOPs in Sect. 2. In Sect. 3, NCIE is introduced. In Sect. 4, our approach will be described in detail. Section 5 reports the experimental results, in which the behavior of our approach is analyzed and discussed. Finally, Sect. 6 gives the conclusion and points out the future work.

## 2 Conflicting and redundant objectives

### 2.1 Conflicting objectives

Simply, the conflict between two objectives means that the improvement on one objective would deteriorate the other objective. The conflict might be global or local (the range of conflict) (Freitas et al. 2013), and linear or nonlinear (the structure of correlation) (Saxena et al. 2013).

### 2.2 Redundant objectives

If there is no conflict between two objectives, one of them can be viewed as a redundant objective for this MOP. Generally, the redundant objectives in an MOP are defined as the objectives that can be ignored without changing the structure of its original PF (Gal and Hanne 1999).

### 2.3 Reducible MOPs

Many-objective optimization problems (MOPs) with redundant objectives are reducible MOPs, which can be applied objective reduction techniques. If such MOPs can be reduced to MOPs with low-dimensional objectives, existing MOEAs can be used.

The above definition is not strictly mathematical. However, as Brockhoff and Zitzler (2006) mentioned, the existing literature has not clarified two main problems for objective reduction. One is the effect of objective reduction on dominance, and the other is the evaluation of the subset of objectives after reduction.

Examples of minimal MOPs with redundant objectives, where \(x \in [0,1]\)

Case 1 | \(\begin{array}{l} {f_1} = {x_1}\\ {f_2} = 1 - {x_1}\\ {f_3} = 2{f_1}\\ PF: f_1+f_2=1,{f_3} = 2{f_1} \end{array}\) |

Case 2 | \(\begin{array}{l} {f_1} = {x_1}\\ {f_2} = 1 - {x_1}\\ {f_3} = \sin (0.5\pi {f_1})\\ PF: f_1+f_2=1,{f_3} = \sin (0.5\pi {f_1}) \end{array}\) |

Case 3 | \(\begin{array}{l} {f_1} = {x_1}\\ {f_2} = 1 - {x_1}\\ {f_3} = 1\\ PF: f_1+f_2=1,{f_3} = 1 \end{array}\) |

Case 4 | \(\begin{array}{l} {f_1} = {x_1}{x_2}(1 + {x_3}^2)\\ {f_2} = {x_1}(1 - {x_2})(1 + {x_3}^2)\\ {f_3} = \left\{ \begin{array}{l} (1 - {x_1})(1 - {x_2})(1 + {x_3}^2),\quad {x_3} \ne 0\\ 0,\quad {x_3} = 0 \end{array} \right. \\ PF: f_1+f_2=1,{f_3} = 0 \end{array}\) |

In Cases 1 and 2, the linear and nonlinear correlation are both important in objective reduction. However, the majority of the existing approaches employ linear tools to describe all the scenarios (Deb and Saxena 2005), which results in poor performance for nonlinear correlation. Comparing Cases 1 and 2, \(f_3\) is a redundant objective for \(f_1\). In Case 1, \(f_3\) is linearly correlated to \(f_1\), but nonlinearly correlated to \(f_1\) in Case 2. If we use linear tools to evaluate the correlation degree, the obtained conflict degree in Case 2 is smaller than that in Case 1. It is obviously less reasonable. That is the reason why we use NCIE to capture a more general correlation for objective reduction.

## 3 Nonlinear correlation information entropy

## 4 Objective reduction based on nonlinear correlation information entropy

### 4.1 Basic idea

The correlation analysis and objective selection are two key steps in an objective reduction approach. For correlation analysis, a majority of the existing approaches are based on the correlation matrix, which is only used for the linear correlation measure. As NCIE can handle both the linear and nonlinear correlation, we adopt it to measure the correlation in our approach. For objective selection, we abandon those common techniques in the existing approaches (such as PCA and feature selection) and design a straightforward method to select conflicting objectives (explained in Sect. 4.3).

The NCIE-based correlation analysis is based on the non-dominated population in every generation; thus, the conflict between objectives are local rather than global (López Jaimes et al. 2014). As Sect. 3 shows, the conflicts might be local in some cases. Thus, our proposed method could reduce some non-globally redundant objectives but locally-redundant objectives. During the execution of MOEAs, the conflict degree would be updated by the value of NCIE. In short, the basic idea of our approach is to keep the most conflicting objectives and omit the most positively correlated objectives in the NCIE matrix during run time.

### 4.2 Correlation analysis

### 4.3 Objective selection

Example of the modified NCIE matrix on DLTZ5(2,5)

\(f_1\) | \(f_2\) | \(f_3\) | \(f_4\) | \(f_5\) | |
---|---|---|---|---|---|

\(f_1\) | 1.0000 | 0.4959 | 0.4244 | 0.5348 | -0.3552 |

\(f_2\) | 0.4959 | 1.0000 | 0.3972 | 0.4686 | -0.3381 |

\(f_3\) | 0.4244 | 0.3972 | 1.0000 | 0.4765 | -0.4352 |

\(f_4\) | 0.5348 | 0.4686 | 0.4765 | 1.0000 | -0.4488 |

\(f_5\) | -0.3552 | -0.3381 | -0.4352 | -0.4488 | 1.0000 |

\(\sum {\text {NCIE}}<0\) | -0.3552 | -0.3381 | -0.4352 | -0.4488 | -1.5773 |

Our approach is different from the approach that outputs a fixed number of objectives (Deb and Saxena 2005). It selects different numbers of objectives according to the situation of the current population, which is more robust for different problems.

### 4.4 Classification for correlated and non-correlated objectives

### 4.5 Computational complexity

For an \(m\)-objective problem with a solution set of size \(N\) (in most cases, \(N\) is larger than \(m\)), the NCIE matrix calculation in the correlation analysis has \(O(m^2N)\) complexity, and the objective selection has \(O(m^2)\) complexity. Therefore, the total complexity of our method is (\(O(m^2N)\)) per generation.

## 5 Experimental studies

### 5.1 Test problems, metrics, and settings

As the DTLZ problems (Deb et al. 2002b) and WFG3 (Huband et al. 2006) are MOPs with different numbers of objectives, we adopt them as the test problems in our experiments. Among these test problems, DTLZ1-4 are irreducible, and DTLZ5 (Deb and Saxena 2005) and WFG3 are reducible. DTLZ5(\(I\),\(M\)) is an \(M\)-objective problem with \(I\) conflicting objectives.

### 5.2 Experiments on analysis of components

#### 5.2.1 Characteristics of DTLZ5

#### 5.2.2 Correlation analysis

The modified NCIE matrix plays an important role in the correlation analysis of our approach. The correlation matrix is another popular metric of correlation of multiple variables. We embed these two different matrices separately into our objective reduction approach to show the behavior of the modified NCIE matrix. The two objective reduction approaches are both embedded in NSGA-II. The differences in reduction performance and execution time are summarized below.

From the above results, we find that the approach based on the modified NCIE matrix performs better than the approach based on the correlation matrix for the problems with a large number (e.g., 50) of objectives, because the correlation matrix cannot provide clear correlation information for the subsequent objective selection.

#### 5.2.3 Classification for objectives

The classification of objectives plays an important role in our approach. We compare our approaches with different \(T\)s and the classification method. The experiment is conducted on the reducible and irreducible problems (DLTZ5(2,10) and DLTZ2 with 10 objectives). Our objective reduction approach is embedded in NSGA-II. For the reducible problems, our objective reduction approach aims to reduce the most redundant objectives. For the irreducible problems, our objective reduction approach aims to keep the right correlation of objectives. Therefore, we adopt the median number of objectives after reduction over 200 generations to evaluate the behavior of our approach.

#### 5.2.4 Objective selection

We adopt a direct method to select the most conflicting objectives according to the obtained NCIE matrix from correlation analysis. The objective reduction approach based on PCA (Saxena et al. 2013) is a well-known one. Therefore, we compare our objective selection method with the PCA method (using the same setting as in Saxena et al. 2013). As NCIE is a nonlinear metric, we also compare it with the kernel PCA (KPCA) method (with Gaussian kernel function). DTLZ5(2,\(M\)) (\(M=5,10,20,30,50\)) is chosen as the test problem in this subsection. We embed all the approaches in NSGA-II. The differences in reduction performance and execution time are summarized below.

Our proposed approach can reduce objectives more efficiently than the approach based on PCA. This is reflected through two aspects, one is the number of objectives after reduction, and the other is the selected objectives.

We find that the objective selection in our approach performs better than the approach based on PCA in different aspects (objective reduction performance and execution time). The main reason why our objective selection approach outperforms the approach based on PCA is that our approach reduces more objectives than the approach based on PCA. Because of the larger number of objectives obtained by the approach based on PCA, it cannot reduce the difficulty of the original problem.

KPCA, as a nonlinear method, maps the data to a high-dimensional space to keep its nonlinear characteristic. However, its kernel function has to be chosen in advance, which affects its performance significantly. From the results, we can find that the Gaussian kernel function is not suitable for DTLZ5.

#### 5.2.5 Population size

### 5.3 Experiments on performance

In this subsection, we apply our approach to both Pareto-based and indicator-based MOEAs (NSGA-II Deb et al. 2002a and IBEA (\(I_{\varepsilon +}\)-based) Zitzler and Künzli 2004). Both the reducible and irreducible problems are tested in the following experiments.

#### 5.3.1 Pareto-based MOEAs

We embed our NCIE-based approach in NSGA-II on DTLZ1-5. The results are analyzed by Mann–Whitney \(U\) test (Hollander and Wolfe 1999). The significant ones are in boldface (the significant level is 0.05).

IGD values of the NSGA-II with our approach and the original NSGA-II on DTLZ5 analyzed by Mann-Whitney \(U\) test

\(I\) | \(M\) | NSGA-II (NCIE) | NSGA-II | \(p\) value |
---|---|---|---|---|

2 | 5 |
| 0.0042 \(\pm \) 0.0000 | 0.0256 |

2 | 10 |
| 1.8023 \(\pm \) 1.5792 | 0.000 |

2 | 20 |
| – | – |

2 | 30 |
| – | – |

2 | 50 |
| – | – |

3 | 5 | 0.1155 \(\pm \) 0.1684 | 0.0550 \(\pm \) 0.0009 | 0.4735 |

3 | 10 |
| 3.4976 \(\pm \) 7.2347 | 0.0000 |

3 | 20 |
| – | – |

5 | 10 |
| 70.7366 \(\pm \) 47.8311 | 0.0003 |

5 | 20 |
| – | – |

7 | 10 | 93.9852 \(\pm \) 58.3535 |
| 0.0315 |

7 | 20 |
| – | – |

IGD values of the NSGA-IIs with our approach and random objectives reduced and the original NSGA-II on DTLZ1-4 analyzed by Mann–Whitney \(U\) test

DTLZ | \(M\) | NSGA-II (NCIE) | NSGA-II (Random) | NSGA-II |
---|---|---|---|---|

1 | 5 |
| 29.7310 \(\pm \) 18.8620 | 28.1565 \(\pm \) 14.7707 |

1 | 15 |
| 36.6916 \(\pm \) 23.5109 | – |

1 | 25 | 17.7863 \(\pm \) 15.8425 | 27.6838 \(\pm \) 21.9433 | – |

2 | 5 | 0.5188 \(\pm \) 0.2108 | 1.2836 \(\pm \) 0.3187 | 0.4893 \(\pm \) 0.0741 |

2 | 15 | 2.1166 \(\pm \) 0.2912 | 2.2857 \(\pm \) 0.2768 | – |

2 | 25 |
| 2.7165 \(\pm \) 0.1811 | – |

3 | 5 | 129.0159 \(\pm \) 62.0312 | 209.6908 \(\pm \) 20.1481 | 160.7862 \(\pm \) 24.2639 |

3 | 15 |
| 232.9653 \(\pm \) 13.5670 | – |

3 | 25 |
| 277.0580 \(\pm \) 107.0687 | – |

4 | 5 | 0.7899 \(\pm \) 0.3023 | 1.1632 \(\pm \) 0.0114 | 0.5651 \(\pm \) 0.0613 |

4 | 15 |
| 1.3412 \(\pm \) 0.0399 | – |

4 | 25 |
| 1.4126 \(\pm \) 0.0750 | – |

#### 5.3.2 Indicator-based MOEAs

IBEA (Zitzler and Künzli 2004) is an indicator-based MOEA, which is well known for its ability for MaOPs. We embed our objective reduction approach in IBEA to analyze its effects on indicator-based MOEAs. The significant results are in boldface after being analyzed by Mann–Whitney \(U\) test (Hollander and Wolfe 1999) (the significant level is 0.05).

IGD values of the IBEA with our approach and the original IBEA on DTLZ5 analyzed by Mann–Whitney \(U\) test

\(I\) | \(M\) | IBEA (NCIE) | IBEA | \(p\) value |
---|---|---|---|---|

2 | 5 | 0.6529 \(\pm \) 0.1694 | 0.5784 \(\pm \) 0.1953 | 0.0764 |

2 | 10 | 0.6324 \(\pm \) 0.1789 |
| 0.0439 |

2 | 20 | 0.6734 \(\pm \) 0.1457 |
| 0.0003 |

2 | 30 | 0.6083 \(\pm \) 0.1835 | 0.5603 \(\pm \) 0.1601 | 0.2503 |

2 | 50 | 0.7018 \(\pm \) 0.1159 | 0.6298 \(\pm \) 0.1997 | 1.0000 |

3 | 5 | 0.7876 \(\pm \) 0.1536 | 0.7506 \(\pm \) 0.1633 | 0.9676 |

3 | 10 | 0.8257 \(\pm \) 0.1953 | 0.7815 \(\pm \) 0.1771 | 0.3369 |

3 | 20 | 0.7874 \(\pm \) 0.2094 | 0.8227 \(\pm \) 0.1836 | 0.9461 |

5 | 10 | 0.9366 \(\pm \) 0.1882 | 0.9312 \(\pm \) 0.2111 | 0.7764 |

5 | 20 | 0.8839 \(\pm \) 0.2212 | 0.8099 \(\pm \) 0.1878 | 0.4903 |

7 | 10 | 1.1193 \(\pm \) 0.1839 | 1.0622 \(\pm \) 0.2286 | 0.8817 |

7 | 20 | 1.1482 \(\pm \) 0.1584 | 1.0911 \(\pm \) 0.2212 | 0.7353 |

IGD values of the IBEA with our approach and random objectives reduced and the original IBEA on DTLZ1-4 analyzed by Mann–Whitney \(U\) test

DTLZ | \(M\) | IBEA (NCIE) | IBEA (Random) | IBEA |
---|---|---|---|---|

1 | 5 | 1.8145 \(\pm \) 3.8061 | 3.7388 \(\pm \) 7.9257 | 3.7992 \(\pm \) 8.2498 |

1 | 15 | 3.9714 \(\pm \) 4.1883 | 3.5229 \(\pm \) 3.9579 | 7.8709 \(\pm \) 10.3937 |

1 | 25 | 13.0244 \(\pm \) 13.0977 | 8.4152 \(\pm \) 12.7078 | 12.6828 \(\pm \) 16.3969 |

2 | 5 | 1.1047 \(\pm \) 0.1011 | 1.1429 \(\pm \) 0.0667 | 1.0741 \(\pm \) 0.2317 |

2 | 15 | 1.3278 \(\pm \) 0.1319 | 1.2825 \(\pm \) 0.0649 | 1.2207 \(\pm \) 0.1260 |

2 | 25 | 1.3969 \(\pm \) 0.1818 | 1.3004 \(\pm \) 0.0977 | 1.3148 \(\pm \) 0.0318 |

3 | 5 | 43.9396 \(\pm \) 37.2782 | 39.3212 \(\pm \) 38.0931 | 34.4497 \(\pm \) 30.4998 |

3 | 15 | 39.3778 \(\pm \) 16.4310 | 42.8500 \(\pm \) 19.0732 | 47.5738 \(\pm \) 30.2017 |

3 | 25 | 39.1774 \(\pm \) 21.4049 | 42.2101 \(\pm \) 20.6839 | 52.6554 \(\pm \) 17.5275 |

4 | 5 | 1.1659 \(\pm \) 0.0002 | 1.1662 \(\pm \) 0.0007 | 1.1658 \(\pm \) 0.0002 |

4 | 15 | 1.3340 \(\pm \) 0.0000 | 1.3340 \(\pm \) 0.0000 | 1.3340 \(\pm \) 0.0000 |

4 | 25 | 1.3652 \(\pm \) 0.0042 | 1.3662 \(\pm \) 0.0001 | 1.3671 \(\pm \) 0.0005 |

The number of objectives appears to have little influence on the behavior of IBEA. In other words, the effect of our approach is not significant on IBEA for either reducible or irreducible problems.

#### 5.3.3 WFG problems

IGD values of the NSGA-II with our approach, the original NSGA-II, the IBEA with our approach, and the original IBEA on WFG3 analyzed by Mann–Whitney \(U\) test

\(M\) | NSGA-II (NCIE) | NSGA-II | \(p\) value | IBEA (NCIE) | IBEA | \(p\) value |
---|---|---|---|---|---|---|

5 |
| 0.4337 \(\pm \) 0.0714 | 0.0000 | 4.5874 \(\pm \) 0.1647 | 4.6532 \(\pm \) 0.0869 | 0.1404 |

10 |
| 2.3687 \(\pm \) 0.4006 | 0.0003 | 9.7115 \(\pm \) 2.2619 | 9.5860 \(\pm \) 2.5905 | 0.9031 |

20 |
| 9.1938 \(\pm \) 0.5134 | 0.0000 | 19.9937 \(\pm \) 5.4725 | 14.3213 \(\pm \) 9.5895 | 0.0859 |

30 |
| – | – | 30.7959 \(\pm \) 8.6535 | 24.3803 \(\pm \) 13.2391 | 0.2393 |

50 |
| – | – | 51.7169 \(\pm \) 14.1403 |
| 0.0275 |

#### 5.3.4 Discussion

Generally, the advantage of IBEA is its good convergence ability on MaOPs, which NSGA-II cannot achieve. However, IBEA cannot obtain the results of good diversity because of the poor performance from \(I_{\varepsilon +}\) (Hadka and Reed 2012). NSGA-II pays more attention to this aspect. Our experimental results support these two points. The aim of our approach is to improve the convergence ability while maintaining the good diversity of NSGA-II.

Our approach works well with Pareto-based MOEAs, but not with indicator-based MOEAs. Pareto-based MOEAs have difficulties on MaOPs because of ineffective Pareto dominance for large numbers of objectives. Our approach can reduce the redundant number of objectives to make the Pareto dominance work again. However, indicator-based MOEAs do not suffer from the Pareto dominance problem, even though the large numbers of objectives decrease their performance too.

For the reducible problems such as DTLZ5 and WFG3, our approach selects the most conflicting objectives, which decreases much computational cost. Thus, the NSGA-II with our approach can solve problems with more objectives, which cannot be solved by the original NSGA-II. For example, DTLZ5(2,50) can be reduced to a two-objective problem, and the computational cost is decreased to 4 % of that for a 50-objective problem. However, with the increasing \(I\) in DTLZ5(\(I\),\(M\)), the NSGA-II with our approach decreases its convergence ability. This is because the difficulties of those problems are still high after objective reduction. For the irreducible problems such as DTLZ1-4, the NSGA-II with our approach can solve those with 25 objectives, whereas the original NSGA-II can only solve the problems with five objectives. Furthermore, the NSGA-II with our approach can also obtain slightly better results than the original NSGA-II, because our approach was able to capture and exploit local objective interactions.

## 6 Conclusion

Since the correlation among redundant objectives might be either linear or nonlinear, the existing linear objective reduction approaches have limitations. We have proposed a novel objective reduction approach based on NCIE, which can handle both linear and nonlinear correlations.

In our approach, we employ NCIE, a nonlinear metric, to measure the correlation among objectives. In addition, we use a simple objective selection method without any pre-defined parameter, which results in the robustness of our approach. The experiments on DTLZ5 in Sect. 5.2.4 shows that our approach can select the most conflicting objectives for reduction. Our approach can be embedded in any MOEA to reduce the number of objectives, as demonstrated by the experiment on NSGA-II and IBEA. The experimental results show that our approach improves Pareto-based MOEAs (NSGA-II) on reducible problems (DTLZ5 and WFG3), but cannot improve the performance of indicator-based MOEAs (IBEA). At the same time, our approach also improves the performance of Pareto-based MOEAs on the irreducible problems (DTLZ1-4) slightly, because the difficulty of the original problems decreases locally, which promotes convergence.

However, there are some disadvantages of the NCIE approach that we have to overcome in our future work. (1) The reduction performance of our approach on the problems with more than 20 objectives is still not ideal. (2) The improvement of our approach on indicator-based MOEAs needs to be strengthened. (3) It will be interesting to evaluate our techniques on hypervolume-based IBEAs. (4) It will be useful to apply our approach to search knee areas of MaOPs (Bechikh et al. 2011).

## Notes

### Acknowledgments

This work was supported by the National Basic Research Program (973 Program) of China (No. 2013CB329402), an EU FP7 IRSES Grant (No. 247619) on “Nature Inspired Computation and its Applications (NICaiA)”, an EPSRC grant (No. EP/J017515/1) on “DAASE: Dynamic Adaptive Automated Software Engineering”, the Program for Cheung Kong Scholars and Innovative Research Team in University (No. IRT1170), the National Natural Science Foundation of China (No. 61329302), National Science Foundation of China under Grant (No.91438103 and 91438201), and the Fund for Foreign Scholars in University Research and Teaching Programs (the 111 Project) (No. B07048). Xin Yao was supported by a Royal Society Wolfson Research Merit Award.

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