Comparison between MOEA/D and NSGA-III on a set of novel many and multi-objective benchmark problems with challenging difficulties

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Abstract

Currently, evolutionary multiobjective optimization (EMO) algorithms have been successfully used to find a good approximation of many-objective optimization problems (MaOPs). To measure the performance of EMO algorithms, many benchmark multiobjective test problems have been constructed. Among them, DTLZ and WFG are two representative test suites with the scalability to the number of variables and objectives. It should be pointed out that MaOPs can be more challenging if they are involved with difficult problem features, such as objective scalability, complicated Pareto set, bias, disconnection, or degeneracy. In this paper, a set of ten new test problems with above-mentioned difficulties are constructed. Some experimental results on these test problems found by two popular EMO algorithms, i.e., MOEA/D and NSGA-III, are reported and analyzed. Moreover, the performance of these two EMO algorithms with different population sizes on these test problems are also studied.

Introduction

Due to the possible conflicting relationship among objectives, the optimality of MOPs is often defined via Pareto dominance. The set of all optimal solutions in the objective space is called Pareto front (PF). In classical multiobjective optimization, a MOP is often converted into a single objective optimization problem via decomposition (also called scalarizition). Classical multiobjective optimization methods, such as weighted sum method and weighted Tchebycheff method, can find only one Pareto solution in one run. In decision-making, the presence of multiple Pareto solutions should be more helpful. Compared with classical methods, EMO algorithms are more powerful in obtaining a population of Pareto solutions in a single run [1]. Fitness assignment and diversity control are two major research issues in EMO algorithms. So far, NSGA-II [2] and MOEA/D [3] have been widely studied in the EMO research area. The former EMO algorithm uses Pareto dominance for fitness assignment while the latter EMO algorithm decomposes a MOP into multiple single objective subproblems. Compared with NSGA-II, MOEA/D has more flexibility in combining existing efficient single objective optimizers. However, NSGA-II requires less effort for parameter settings.

Over the past ten years, a lot of efforts have been made to solve MaOPs [[4], [5], [6], [7], [8], [9]]. Unlike the MOPs with two or three objectives, the PFs of MaOPs are usually manifolds with high dimensionality. Due to this fact, it is difficult to approximate high-dimensional PFs via maintaining a population with limited size in Pareto-based EMO algorithms. This is because the population members in these EMO algorithms are very likely to be nondominated to each other even in the beginning of search. As a result, the population of Pareto-based EMO algorithms may fail to converge towards the PFs of MaOPs. In many existing literature, it has been proved that the pure Pareto-based EMO algorithms like NSGA-II has difficulties in solve MaOPs. In contrast, the EMO algorithms based on decomposition like MOEA/D are very suitable for dealing with MaOPs since they evolve a population of solutions by optimizing multiple single objective subproblems, which could be defined in terms of weight vectors, reference vectors, or subregions. Each subproblem is associated with one search direction towards PF. To achieve satisfactory performance, the diversity of subproblems and the association between subproblems and solutions should be highly addressed in MOEA/D. In recent a few years, the combination of Pareto-based EMO algorithms and decomposition has also been studied. For example, the selection of population for next generation in NSGA-III [6] is done by associating solutions in the nondominated front with a number of prespecified reference points. Note that the role of reference points in NSGA-III is similar with that of search directions (corresponding to subproblems) in MOEA/D.

To measure the performance of EMO algorithms, many benchmark multiobjective test problems [[10], [11], [12]] have been studied. Among them, DTLZ and WFG are two popular multiobjective test suites, which are scalable to the number of objectives. During the past ten years, both DTLZ and WFG test problems have been widely used in comparing the performance of many-objective EMO algorithms. Note that all DTLZ test problems have no parameter dependencies between position-related variables and distance-related variables. The Pareto sets (PSs) of these test problems are linearly distributed in the decision space. It should be mentioned that one Pareto solution in the PFs of DTLZ test problems can be produced by disturbing the position-related variables of other Pareto solutions. To impose the difficulty between position-related variables and distance-related variables, some challenging multiobjective test problems with complicated PSs were studied in [12]. The extension of MaOPs with complicated PSs was also investigated in [13].

It is well-established that the PFs of MOPs are (M − 1)-D manifold under certain mild conditions.1 For example, the PF of ZDT1 is a 1-D curve while the PF of DTLZ1 is a 2-D surface. Approximating the PFs of the bi-objective ZDT1 and the three-objective DTLZ1 is not a difficult task for either Pareto-based EMO algorithms or decomposition-based EMO algorithms when the population with hundreds of solutions is considered. When dealing with many-objective DTLZ test problems, decomposition-based EMO algorithms are still effective for finding a set of well-distributed Pareto solutions along the PF. Usually, decomposition-based EMO algorithms define multiple subproblems via a number of weight vectors or reference points or reference directions with good spread. This decomposition strategy can work well on the MOPs with regular PF shapes like DTLZ1 (simplex) and DTLZ2 (spherical surface), but may fail to solve the MOPs with irregular PF shapes. In fact, the PFs of MaOPs could be degenerate if objectives are positively correlated. Therefore, approximating the degenerate PFs of MaOPs can be a very challenging task for decomposition-based EMO algorithms.

Apart from complicated PSs and degeneracy, some other challenging problem difficulties in multi-objective optimization, such as objective scalability, multi-modality, disconnectedness, and bias [14], have not yet widely been studied in many-objective optimization. In [15], we constructed a set of ten new challenging benchmark test problems with above-mentioned difficulties. To verify the difficulties of these test problems, two well-known EMO algorithms, i.e., MOEA/D and NSGA-III, are considered for performance comparison in this paper. It should be pointed out that the majority of existing research work on MOEA/D and NSGA-III for many-objective optimization considers the use of small population size. From the decision-making point of view, this is a good strategy for population setting since each solution is associated with one prior reference point. In fact, the use of large population size in EMO algorithms should be encouraged if the set of reference points are not available. In this work, the experimental results on MOEA/D and NSGA-III for our proposed test problems are also reported and analyzed.

The rest of this paper is organized as follows. Section 2 gives a brief review on some commonly-used multiobjective test problems and presents the detailed formulations of ten new test problems with challenging difficulties. Two baseline EMO algorithms, i.e. MOEA/D and NSGA-III, are illustrated in Section 3. To study the difficulties of our proposed test problems, some experimental results are reported and discussed in Section 4. The final section concludes the paper.

Section snippets

Challenging test problems for many and multi-objective optimization

The construction of benchmark multiobjective test problems is one of the important research issues in the EMO research area. The studies of characteristics and difficulties in these benchmark test problems will assist the development of efficient EMO algorithms. Table 1 summarizes the characteristics of several commonly-used benchmark multiobjective test problems. The difficulties of these characteristics are introduced as follows:

  • ZDT [16]: The concavity in ZDT2 can cause difficulty for the EMO

Two baseline EMO algorithms: MOEA/D and NSGA-III

MOEA/D was first proposed in [3]. Unlike Pareto-based EMO algorithms, the main goal of MOEA/D is to optimize multiple single objective subproblems or multiple small multi-objective optimization problems. To optimize multiple subproblems, a number of weight vectors with good distribution are often needed. During the optimization process of MOEA/D, the neighborhood relationships among subproblems are used in selecting mating parents and population replacement. Since MOEA/D provides a very general

Computational experiments

In this section, the experimental results of MOEA/D and NSGA-III2 on the proposed test instances are reported and analyzed. The performance of both algorithms with different settings of population size is also studied.

Conclusions

In this paper, we have suggested a set of ten novel multi/many-objective test problems, which involve some challenging difficulties in multiobjective optimization, such as objective scalability, complicated PS, bias in diversity or convergence, disconnection, and degeneracy. Some experimental results on MOEA/D and NSGA-III for our new test problems with three objectives or five objectives have been conducted. We found that both algorithms have difficulty in approximating the PFs of MaOP1-MaOP10

Acknowledgment

The authors would like to thank the anonymous reviewers for their insightful comments. This work was supported by National Natural Science Foundation of China under Grant 61573279, Grant 61175063, Grant U1811461, Grant 11690011, and Grant 61721002. This work was also supported by a grant from ANR/RGC Joint Research Scheme sponsored by the Research Grants Council of the Hong Kong Special Administrative Region, China and France National Research Agency (Project No. A-CityU101/16).

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