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Decomposition-based multiobjective evolutionary algorithms (MOEA/Ds) have become increasingly popular in recent years. In these MOEA/Ds, evolutionary search is guided by the used weight vectors in decomposition function to approximate the Pareto front (PF). Generally, the decomposition function will be constructed by the weight vectors and the reference point, which play an important role to balance convergence and diversity during the evolutionary search. However, in most existing MOEA/Ds, only one ideal point is used as the reference point for all the evolutionary search, which is harmful to search the entire PF when tackling the problems with difficult-to-approximate PF boundaries. To address the above problem, this paper proposes an evolutionary search method with multiple utopian reference points in MOEA/Ds. Similar to the existing MOEA/Ds, each solution is associated with one weight vector, which provides an evolutionary search direction, while the novelty of our approach is to use multiple utopian reference points, which can provide evolutionary search directions for different weight vectors. Corner solutions are used to approximate the nadir point and then multiple utopian reference points for evolutionary search can be constructed based on the ideal point and the nadir point, which are uniformly distributed on the coordinate axis or planes. The use of these utopian points can prevent solutions to gather in the same region of PF and helps to strike a good balance of exploration and exploitation in the search space. The performance of our proposed algorithm is validated on tackling 16 recently proposed test problems with difficult-to-approximate PF boundaries and empirically compared to eight state-of-the-art multiobjective evolutionary algorithms. The experimental results demonstrate the superiority of the proposed algorithm on solving most of the test problems adopted.

In some real-world applications [

Most of MOEAs can be classified to three categories based on the selection mechanisms:

One popular decomposition-based framework for MOEA (MOEA/D) proposed by Zhang_{r} individuals in its neighborhood to prevent diversity loss, where_{r} is preset as a small positive integer. For example, in MOEA/D-STM [

However, it is found in [

The rest of this paper is organized as follows. Section

In decomposition approaches, the scalar subproblem is formulated as a weighted aggregation of all the objectives by decomposition function. The TCH decomposition [^{−6}, in case

Let _{1}-norm.

Tchebycheff decomposition with the ideal reference point for the cases: (a)_{1}. (b)_{1}.

In the case that_{1}, the following proposition depicts the geometric property of

Let _{1} as shown in Figure

The above two propositions demonstrate the geometric property of TCH decomposition in a serious mathematical proof. More details of proofs have been systematically discussed and analyzed in [

There have been numerous studies of decomposition approaches [

The ideal point

The nadir point

The utopian point

As shown in Figure

Four different cases of reference points in decomposition functions: (a) the ideal point; (b) the nadir point; (c) the utopian point; (d) multiple reference points.

In decomposition approaches, each subproblem is built by using a weight vector and a reference point, which provides an evolutionary search direction for its associated solution. As it is impossible to know the exact ideal point beforehand, an approximate ideal point is often used as the reference point for all the search directions in MOEA/Ds [

The missing of boundary PF parts in the cases: (a) the ideal reference point: (b) the nadir reference point.

Therefore, for the problems with difficult-to-approximate PF boundaries, this paper proposes an evolutionary search method with multiple utopian reference points (MUP) in MOEA/Ds, which can avoid overcrowdedness at the middle part of the PF and search the entire PF. Similar to the existing MOEA/Ds, each solution is associated with one weight vector, which can provide an evolutionary search direction. However, the main difference of our approach with the existing MOEA/Ds is that multiple utopian reference points are used to provide evolutionary search directions for different weight vectors. As these utopian reference points are uniformly distributed along all the coordinate axis or planes that are constructed by the ideal reference point and the nadir reference point, a much wider breadth-diversity [

As recently revealed in [

In this section, the generation of multiple utopian points (MUP) is first introduced in Section

In existing MOEA/Ds [

Illustration of a utopian point, narrow utopian area, and dominated area.

In this paper, one more generalized formula of the utopian point is given, as mathematically defined by

In our proposed decomposition method with MUP, the weight vectors and the reference points are still the critical components. Given a set of uniformly distributed weight vectors

The ideal point and the nadir point decide the range of these newly generated utopian points as illustrated in (

In our proposed method, each utopian point is associated with a weight vector as illustrated in (

An illustration of correlation between a utopian point and its corresponding search direction for two objectives.

In our proposed decomposition approach, a set of uniformly distributed utopian reference points replaces the ideal reference point to guide the search behaviors of individuals. Due to the replacement of reference point, the classical TCH decomposition function in (

Without using the absolute operator in (

Let

Tchebycheff decomposition with the utopian reference point.

In the above equation, since

Let

In this proposition, there are two different cases according to the dominance relation between the solution and the utopian reference point in the objective space. As shown in Figure

According to the conditions of

In geometry, _{1}-norm of _{2} and the contour line of

In this section, we introduce the framework of our evolutionary search method with MUP in MOEA/Ds, i.e., MOEA/D-MUP, whose pseudocode is given in Algorithm

^{ide}: the ideal point

^{nad}: the nadir point

Sixteen test problems [

Three classical MOEAs (NSGA-II [

As shown in Table

The population size:

Reproduction operators: To be fair, we use DE operator and Polynomial-based mutation for offspring generation in all the compared algorithms except that MOEA/D-M2M employs another recombination operator [

The stopping condition and the number of runs: The maximal number of function evaluation is set to 300,000 for all test problems. All the algorithms are run 30 times independently on each test problem.

Parameter settings of all the compared algorithms.

Algorithm | Parameter settings |
---|---|

NGSA-II | |

SMS-EMOA | |

MOEA/D-DRA | |

MOEA/D-DRA-UT | |

MOEA/D-STM | |

MOEA/D-M2M | |

MOEA/D-AWA | |

MOEA/D-PaS | |

MOEA/D-MUP | |

To compare the performance of all algorithms on test instances, the inverted generational distance (IGD) indicator and the hypervolume indicator (_{H}) are used in our experiments. A set of solutions uniformly sampled along the true PF is needed for computing the former indicator, i.e., IGD. However, for the later, i.e.,_{H}, a reference point is necessary. In our experiment, all the nondominated solutions are selected from the final population, which are used to compare the performance of all algorithms in terms of IGD values and_{H} values.

_{ H } [_{H} value is computed as follows:

In this section, we present and discuss the comparisons of MOEA/D-MUP with other state-of-the-art algorithms. Three classical algorithms (NSGA-II, SMS-MOEA, and MOEA/D-DRA), and a competitive algorithm with the utopian reference point (MOEA/D-DRA-UT) are chosen for comparison. Experimental results obtained by the above four algorithms in terms of the two common indicators, i.e., IGD and_{H}, are shown in Tables _{H} obtained by the four MOEA/D variants are shown in Tables

Mean and standard deviation of IGD values obtained by four classical algorithms and MOEA/D-MUP.

Problem | NSGA-II | SMS-EMOA | MOEA/D-DRA | MOEA/D-DRA-UT | MOEA/D-MUP |
---|---|---|---|---|---|

F1 | 2.73E-02 (3.07E-03)- | 3.43E-02 (5.81E-03)- | 1.88E-02 (1.59E-03)- | 3.93E-03 (3.07E-03)- | |

F2 | 3.28E-02 (2.75E-03)- | 4.47E-02 (3.22E-03)- | 2.90E-02 (1.86E-03)- | 4.02E-03 (2.75E-03)- | |

F3 | 4.84E-02 (7.66E-03)- | 6.39E-02 (9.64E-03)- | 3.80E-02 (3.39E-03)- | 5.09E-03 (7.66E-03)- | |

F4 | 5.73E-02 (6.22E-03)- | 7.50E-02 (5.05E-03)- | 4.48E-02 (2.94E-03)- | 5.21E-03 (6.22E-03)- | |

F5 | 6.35E-02 (2.13E-02)- | 8.43E-02 (2.54E-02)- | 6.34E-02 (1.22E-02)- | 5.14E-03 (2.13E-02)- | |

F6 | 8.19E-02 (1.99E-02)- | 1.13E-01 (1.55E-02)- | 7.10E-02 (8.72E-03)- | 6.19E-03 (1.99E-02)- | |

F7 | 1.92E-01 (1.07E-02)- | 1.77E-01 (6.64E-03)- | | | 1.71E-01 (9.01E-03) |

F8 | 5.86E-02 (1.60E-02)- | 9.17E-02 (1.65E-02)- | 6.32E-02 (2.26E-03)- | | 8.41E-03 (2.91E-03) |

F9 | 1.35E-01 (2.27E-02)- | 1.02E-01 (7.36E-03)- | 5.42E-02 (3.92E-02)- | 4.97E-02 (2.27E-02)- | |

F10 | 1.29E-01 (2.31E-02)- | 1.02E-01 (3.36E-03)- | 5.66E-02 (3.54E-02)- | 5.62E-02 (2.31E-02)- | |

F11 | 1.60E-01 (3.49E-02)- | 1.16E-01 (3.48E-03)- | 6.89E-02 (5.42E-02)- | 6.17E-02 (3.49E-02)- | |

F12 | 1.65E-01 (2.51E-02)- | 1.18E-01 (3.88E-03)- | 7.40E-02 (4.19E-02)- | 6.30E-02 (2.51E-02)- | |

F13 | 2.03E-01 (5.73E-02)- | 1.36E-01 (1.27E-02)- | 9.21E-02 (7.91E-02)- | 7.32E-02 (5.73E-02)- | |

F14 | 1.88E-01 (2.89E-02)- | 1.46E-01 (6.56E-03)- | 9.80E-02 (3.27E-02)- | 7.59E-02 (2.89E-02)- | |

F15 | 2.18E-01 (1.57E-02)- | 1.99E-01 (3.88E-03)- | 1.40E-01 (1.40E-02)- | | |

F16 | 5.35E-02 (9.17E-03)- | 7.61E-02 (6.07E-03)- | 6.58E-02 (7.53E-03)- | 4.23E-02 (9.17E-03)- | |

+/-/= | 0/16/0 | 0/16/0 | 0/15/1 | 2/13/1 | ∖ |

According to Wilcoxon’s rank sum test with a significant level

Mean and standard deviation of _{H} values obtained by four classical algorithms and MOEA/D-MUP.

Problem | NSGA-II | SMS-EMOA | MOEA/D-DRA | MOEA/D-DRA-UT | MOEA/D-MUP |
---|---|---|---|---|---|

F1 | 4.70E-01 (1.59E-03)- | 4.65E-01 (3.24E-03)- | 4.80E-01 (3.07E-03)- | 4.93E-01 (1.02E-03)- | |

F2 | 4.66E-01 (1.86E-03)- | 4.58E-01 (1.63E-03)- | 4.71E-01 (2.75E-03)- | 4.93E-01 (7.69E-04)- | |

F3 | 4.50E-01 (3.39E-03)- | 4.39E-01 (4.60E-03)- | 4.64E-01 (7.66E-03)- | 4.91E-01 (2.66E-03)- | |

F4 | 4.45E-01 (2.94E-03)- | 4.31E-01 (2.38E-03)- | 4.57E-01 (6.22E-03)- | 4.91E-01 (2.38E-03)- | |

F5 | 4.31E-01 (1.22E-02)- | 4.14E-01 (1.21E-02)- | 4.42E-01 (2.13E-02)- | 4.91E-01 (1.28E-03)- | |

F6 | 4.19E-01 (8.72E-03)- | 3.98E-01 (6.95E-03)- | 4.36E-01 (1.99E-02)- | | |

F7 | 3.27E-01 (4.10E-03)- | 3.29E-01 (2.78E-03)- | 3.35E-01 (1.07E-02)- | | 3.56E-01 (4.01E-03) |

F8 | 7.93E-01 (2.26E-03)- | 7.81E-01 (2.23E-03)- | 7.83E-01 (1.60E-02)- | | 8.24E-01 (3.87E-03) |

F9 | 5.72E-01 (3.92E-02)- | 6.16E-01 (2.45E-02)- | 6.76E-01 (2.27E-02)- | | 7.38E-01 (7.61E-03) |

F10 | 5.60E-01 (3.54E-02)- | 5.97E-01 (1.08E-02)- | 6.70E-01 (2.31E-02)- | 6.99E-01 (1.08E-02)- | |

F11 | 5.21E-01 (5.42E-02)- | 5.71E-01 (3.13E-02)- | 6.34E-01 (3.49E-02)- | 6.92E-01 (2.56E-02)- | |

F12 | 5.04E-01 (4.19E-02)- | 5.57E-01 (1.89E-02)- | 6.22E-01 (2.51E-02)- | 6.73E-01 (1.42E-02)- | |

F13 | 4.54E-01 (7.91E-02)- | 5.37E-01 (6.65E-02)- | 5.80E-01 (5.73E-02)- | | |

F14 | 4.58E-01 (3.27E-02)- | 5.02E-01 (3.22E-02)- | 5.69E-01 (2.89E-02)- | | |

F15 | 4.13E-01 (1.40E-02)- | 4.17E-01 (3.52E-03)- | 4.75E-01 (1.57E-02)- | | 4.90E-01 (4.29E-03) |

F16 | 9.18E-01 (7.53E-03)- | 4.65E-01 (3.24E-03)- | 9.27E-01 (9.17E-03)- | 9.51E-01 (3.97E-03)- | |

+/-/= | 0/16/0 | 0/16/0 | 0/16/0 | 1/9/6 | ∖ |

According to Wilcoxon’s rank sum test with a significant level

Mean and standard deviation of IGD values obtained by four representative MOEA/Ds and MOEA/D-MUP.

Problem | MOEA/D-STM | MOEA/D-M2M | MOEA/D-AWA | MOEA/D-PaS | MOEA/D-MUP |
---|---|---|---|---|---|

F1 | 1.47E-02 (4.39E-03)- | 8.58E-03 (2.01E-03)- | 8.43E-02 (2.18E-03)- | 1.74E-02 (2.19E-03)- | |

F2 | 2.14E-02 (4.00E-03)- | 1.25E-02 (2.80E-03)- | 8.95E-02 (2.12E-03)- | 2.07E-02 (2.09E-03)- | |

F3 | 2.80E-02 (7.22E-03)- | 1.07E-02 (2.18E-03)- | 1.47E-01 (3.64E-03)- | 3.12E-02 (3.03E-03)- | |

F4 | 3.81E-02 (6.41E-03)- | 1.41E-02 (3.25E-03)- | 1.54E-01 (2.54E-03)- | 3.59E-02 (2.67E-03)- | |

F5 | 2.61E-02 (1.95E-02)- | 1.52E-02 (4.28E-03)- | 2.29E-01 (3.30E-03)- | 7.30E-02 (7.81E-02)- | |

F6 | 5.16E-02 (1.96E-02)- | 1.75E-02 (4.56E-03)- | 2.37E-01 (3.00E-03)- | 1.39E-01 (1.37E-01)- | |

F7 | | | 2.54E-01 (1.90E-02)- | 2.02E-01 (6.36E-02)- | 1.71E-01 (9.01E-03) |

F8 | 4.41E-02 (9.46E-03)- | 2.16E-02 (6.39E-03)- | 1.90E-01 (2.92E-03)- | 5.84E-02 (5.55E-02)- | |

F9 | 4.51E-02 (2.60E-03)- | 7.07E-02 (2.40E-03)- | 4.96E-02 (2.15E-03)- | 4.18E-02 (2.42E-03)- | |

F10 | 4.75E-02 (1.47E-03)- | 7.39E-02 (2.33E-03)- | 5.17E-02 (2.22E-03)- | 4.52E-02 (3.95E-03)- | |

F11 | 6.05E-02 (6.23E-03)- | 8.91E-02 (5.95E-03)- | 6.91E-02 (3.65E-03)- | 5.27E-02 (4.14E-03)- | |

F12 | 6.49E-02 (2.65E-03)- | 9.83E-02 (1.03E-02)- | 7.25E-02 (2.49E-03)- | 5.96E-02 (5.30E-03)- | |

F13 | 7.92E-02 (1.28E-02)- | 9.35E-02 (1.04E-02)- | 1.06E-01 (3.93E-03)- | 7.12E-02 (7.55E-03)- | |

F14 | 8.45E-02 (1.02E-02)- | 9.93E-02 (1.65E-02)- | 1.13E-01 (5.05E-03)- | 8.06E-02 (2.07E-02)- | |

F15 | 1.40E-01 (3.81E-03)- | 2.14E-01 (5.36E-03)- | 1.72E-01 (8.14E-03)- | 1.45E-01 (5.10E-03)- | |

F16 | 7.10E-02 (5.00E-03)- | 5.16E-02 (2.78E-03)- | 8.78E-02 (6.21E-03)- | 6.85E-02 (4.07E-03)- | |

+/-/= | 0/15/1 | 1/15/0 | 0/16/0 | 0/16/0 | ∖ |

According to Wilcoxon’s rank sum test with a significant level

Mean and standard deviation of _{H} values obtained by four representative MOEA/Ds and MOEA/D-MUP.

Problem | MOEA/D-STM | MOEA/D-M2M | MOEA/D-AWA | MOEA/D-PaS | MOEA/D-MUP |
---|---|---|---|---|---|

F1 | 4.84E-01 (4.07E-03)- | 4.86E-01 (3.43E-03)- | 4.35E-01 (1.19E-03)- | 4.80E-01 (2.94E-03)- | |

F2 | 4.78E-01 (3.77E-03)- | 4.80E-01 (4.69E-03)- | 4.32E-01 (1.26E-03)- | 4.77E-01 (2.22E-03)- | |

F3 | 4.71E-01 (6.77E-03)- | 4.81E-01 (3.98E-03)- | 3.90E-01 (1.77E-03)- | 4.67E-01 (3.34E-03)- | |

F4 | 4.63E-01 (5.40E-03)- | 4.76E-01 (5.16E-03)- | 3.86E-01 (1.57E-03)- | 4.63E-01 (3.18E-03)- | |

F5 | 4.73E-01 (1.80E-02)- | 4.73E-01 (8.50E-03)- | 3.34E-01 (1.80E-03)- | 4.32E-01 (6.19E-02)- | |

F6 | 4.51E-01 (1.74E-02)- | 4.69E-01 (8.67E-03)- | 3.28E-01 (1.68E-03)- | 3.78E-01 (1.10E-01)- | |

F7 | 3.49E-01 (7.79E-03)- | | 3.04E-01 (1.14E-02)- | 3.21E-01 (5.39E-02)- | 3.56E-01 (4.01E-03) |

F8 | 7.97E-01 (5.39E-03)- | 8.10E-01 (5.14E-03)- | 7.33E-01 (1.40E-03)- | 7.78E-01 (7.00E-02)- | |

F9 | 6.83E-01 (1.98E-02)- | 6.60E-01 (1.36E-02)- | 6.99E-01 (3.22E-03)- | 7.09E-01 (1.37E-02)- | |

F10 | 6.67E-01 (4.42E-03)- | 6.45E-01 (6.63E-03)- | 6.91E-01 (2.99E-03)- | 6.92E-01 (6.82E-03)- | |

F11 | 6.41E-01 (3.00E-02)- | 6.39E-01 (2.69E-02)- | 6.47E-01 (4.72E-03)- | 6.72E-01 (1.22E-02)- | |

F12 | 6.23E-01 (6.72E-03)- | 6.23E-01 (2.90E-02)- | 6.39E-01 (3.32E-03)- | 6.61E-01 (1.42E-02)- | |

F13 | 6.06E-01 (5.73E-02)- | | 5.74E-01 (2.90E-03)- | 6.28E-01 (2.39E-02)- | |

F14 | 5.88E-01 (4.53E-02)- | | 5.65E-01 (3.76E-03)- | 6.15E-01 (1.35E-02)- | |

F15 | 4.81E-01 (6.37E-03)- | 4.03E-01 (8.49E-03)- | 4.56E-01 (9.47E-03)- | 4.84E-01 (5.15E-03)- | |

F16 | 9.16E-01 (3.50E-03)- | 9.26E-01 (2.18E-03)- | 9.36E-01 (1.35E-03)- | 9.31E-01 (2.83E-03)- | |

+/-/= | 0/16/0 | 1/13/2 | 0/16/0 | 0/16/0 | ∖ |

According to Wilcoxon’s rank sum test with a significant level of 0.05, the experimental results obtained by the nine MOEAs in terms of IGD metric and_{H} metric are shown in Tables _{H} metric on F7. For F8, MOEA/D-MUP only performs slightly worse than MOEA/D-DRA-UT in terms of mean IGD metric, while they have no significant difference in terms of mean_{H} metric. Next, we specifically analyze and discuss the superiority and drawback of MOEA/D-MUP with four classical MOEAs and four representative MOEA/Ds, respectively. The comparison results of MOEA/D-MUP with four classical MOEAs in terms of mean IGD and_{H} values are presented in Tables

As shown in Tables _{H} values on all test problems adopted, i.e., F1-F16. In fact, the fast nondominated sorting approach always tends to choose these solutions in the middle part of the PF, which is caused by the characteristics of the problems that the solutions located in the middle part of the PF always can dominate other boundary solutions during the evolutionary process. Thus, as shown in Tables _{H} values, MOEA/D-MUP achieves better performance than MOEA/D-DRA on all test problems except that they have no significant difference on F7 regarding the mean IGD value. For MOEA/D-DRA-UT, more boundary areas can be found in the evolutionary search process, while MOEA/D-DRA-UT is still beaten by our proposed MOEA/D-DRA-UT on most test problems. These results validate that solutions in the middle part of the PF are likely maintained in the environmental selection mechanisms of these traditional MOEAs when tackling the problems with difficult-to-approximate PF boundaries. It can be observed from Tables _{H} values. However, on F7, no algorithm can approximate the entire PF due to the extreme difficulty to search the PF boundaries. Quantitatively, MOEA/D-MUP is slightly beaten by MOEA/D-DRA-UT in terms of mean IGD and_{H} values on F7. In addition, the performance of MOEA/D-DRA is similar to MOEA/D-MUP in terms of IGD on F7. This is mainly because the used dynamic resource allocation strategy can allocate more computational resources to search the undiscovered boundaries of the true PF on F7. Quantitatively, for F8, MOEA/D-MUP is slightly beaten by MOEA/D-DRA-UT in terms of IGD metric, while they have similar results in terms of_{H} metric. As discussed above, the use of the utopian reference point instead of the ideal reference point is helpful to exploit the boundaries of the PF. However, in MOEA/D-MUP, multiple utopian reference points are used to provide different evolutionary search directions, which fails to cover the entire PF of F8. This is mainly because the inaccuracy of the nadir point leads to the wrong judgment about the upper bounds of the entire PF of F8. Furthermore, For F9-F16, MOEA/D-MUP achieves a better performance than MOEA/D-DRA-UT in terms of mean IGD metric, except that they are similar on F15. In terms of mean_{H} metric, MOEA/D-MUP performs much better than MOEA/D-DRA-UT on F10-F12 and F16, and they obtain statistically similar results for the rest three-objective test problems. In general, these results confirm the superiority of MOEA/D-MUP for tackling these problems with difficult-to-approximate PF boundaries.

As shown in Tables _{H}. On other problems, MOEA/D-MUP is significantly better than MOEA/D-M2M. In terms of IGD, MOEA/D-STM is similar to MOEA/D-MUP on F7, while MOEA/D-STM is significantly worse than MOEA/D-MUP on the rest problems. When compared to MOEA/D-AWA and MOEA/D-PaS, MOEA/D-MUP performs much better on all test problems. Furthermore, when compared to MOEA/D-STM, MOEA/D-M2M, MOEA/D-AWA, and MOEA/D-PaS, MOEA/D-M2M attains the best mean IGD and_{H} values on all two-objective test problems. From these comparison results, it can be concluded that our proposed MOEA/D-MUP obtains the best performance, and the decomposition strategy used in MOEA/D-M2M achieves some success for some problems with difficult-to-approximate PF boundaries, as it can effectively maintain the boundary solutions into the next population. To conclude, our proposed MOEA/D-MUP can attain significantly better convergence and breadth-diversity than other compared MOEAs on most test problems adopted. For the problems with difficult-to-approximate PF boundaries, diversity should be more considered during the evolution.

To visually show the convergence speeds of all the compared algorithms during the evolution, the curves of IGD value over the number of evaluations are depicted in Figure

The convergence curves of all the compared algorithms on (a) F1, (b) F6, (c) F10, (d) F13, and (e) F16.

The final populations obtained by MOEA/D-DRA-UT, MOEA/D-M2M, and MOEA/D-MUP on F3, F5, F6, and F8.

The final populations obtained by MOEA/D-DRA-UT, MOEA/D-M2M, and MOEA/D-MUP on F9, F12, and F16.

Due to page limitations, we select two competitive algorithms, i.e., MOEA/D-DRA-UT and MOEA/D-M2M, to qualitatively demonstrate the superiority of our proposed algorithm on solving most of the test problems adopted. For example, some representative two-objective problems, i.e., F1, F3, F5, F6, and F8 are shown in Figure

As shown in Figure

In our algorithm, the maximal number of solutions replaced by a newly generated solution could be set as large as

Parameter sensitivity studies of

In this paper, we have proposed an evolutionary search method with multiple utopian reference points for MOEA/D, called MOEA/D-MUP, which uses multiple utopian reference points to provide evolutionary search directions for different weight vectors. In order to effectively exploit the undiscovered boundaries of PF and enhance the breadth-diversity, the utopian reference point of each search direction is got by transforming the ideal point at a specified distance. Moreover, the nadir point is used to restrict the maximal translational distance. In other words, the positions of multiple utopian reference points in the objective space are bounded by the ideal point and the nadir point. Comparing our proposed algorithm with eight state-of-the-art MOEAs in terms of both IGD and_{H} metrics, we have witnessed the effectiveness and competitiveness of MOEA/D-MUP on F1-F16 used in this paper. As indicated by the experimental results, most existing MOEA/Ds cannot approximate the entire PFs on these test problems because the boundaries of PF cannot be effectively exploited. For the problems with difficult-to-approximate PF boundaries, there is a little effort that has been made to

In the future work, we would like to investigate more effective selection mechanisms to better balance convergence and diversity during the evolutionary search process when tackling the problems with difficult-to-approximate PF boundaries. Specifically, we will further study a more accurate generation method for utopian reference points and design recombination operators with stronger search ability. We also will put more efforts to develop mechanisms that can adaptively approximate the complex PFs and to extend our algorithm for solving many-objective optimization problems. Moreover, it is valuable to apply our proposed algorithm to solve some real-world applications [

All the source code and data can be provided by contacting the corresponding author.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China under Grants 61876110, 61836005, and 61672358, the Joint Funds of the National Natural Science Foundation of China under Key Program Grant U1713212, the Natural Science Foundation of Guangdong Province under Grant 2017A030313338, and Shenzhen Technology Plan under Grant JCYJ20170817102218122. Also, this work was supported by the National Engineering Laboratory for Big Data System Computing Technology.