Applying graph-based differential grouping for multiobjective large-scale optimization

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Abstract

An increasing number of multiobjective large-scale optimization problems (MOLSOPs) are emerging. Optimization based on variable grouping and cooperative coevolution is a good way to address MOLSOPs, but few attempts have been made to decompose the variables in MOLSOPs. In this paper, we propose multiobjective graph-based differential grouping with shift (mogDG-shift) to decompose the large number of variables in an MOLSOP. We analyze the variable properties, then detect the interactions among variables, and finally group the variables based on their properties and interactions. We modify the decision variable analyses (DVA) in the multiobjective evolutionary algorithm based on decision variable analyses (MOEA/DVA), extend graph-based differential grouping (gDG) to MOLSOPs, and test the method on many MOLSOPs. The experimental results show that mogDG-shift can achieve 100% grouping accuracy for LSMOP and DTLZ as well as almost all WFG instances, which are much better than DVA. We further combine mogDG-shift with two representative multiobjective evolutionary algorithms: the multiobjective evolutionary algorithm based on decomposition (MOEA/D) and the non-dominated sorting genetic algorithm II (NSGA-II). Compared with the original algorithms, the algorithms combined with mogDG-shift show improved optimization performance.

Introduction

In many real-world problems, it is necessary to consider various objectives simultaneously. Such problems can be treated as multiobjective optimization problems (MOPs). In an MOP, the various objectives always conflict with each other, and a set of possible solutions are required. Because of their ability to generate a set of solutions and their insensitivity to the shape of the Pareto front (PF), multiobjective evolutionary algorithms (MOEAs) [1,2] are capable of solving MOPs, and consequently, MOEAs have been applied to fields such as financial problems [3], satellite communications [4], industrial problems [5,6], data mining [7,8], etc.

With the advent of “big data”, the dimensionality of MOPs is increasing. Many real-world problems can be treated as multiobjective large-scale optimization problems (MOLSOPs), such as the multiobjective large-scale recommendation problem [9]. An MOLSOP includes numerous variables, which challenges traditional MOEAs, such as the multiobjective evolutionary algorithm based on decomposition (MOEA/D) [10] and non-dominated sorting genetic algorithm II (NSGA-II) [11]. Several studies conducted on large-scale global optimization problems (LSGOPs) using approaches that allocate variables to multiple groups and optimize them under the cooperative coevolutionary (CC) [12,13] framework have shown good results. In this paper, to facilitate the more effective optimization of MOLSOPs, we focus on applying graph-based differential grouping (gDG) [14] to these large-scale problems utilizing the CC framework which is conducive to resolving the big data problem.

Additionally, in MOPs, the variable properties [15,16] should be considered, which is not an issue in global optimization problems (GOPs). Position variables affect the diversity of the solutions, distance variables control the convergence of the solutions, while mixed variables affect both diversity and convergence. In the multiobjective evolutionary algorithm based on decision variable analyses (MOEA/DVA) [16], a grouping method for MOLSOPs called decision variable analyses (DVA) has been proposed. In DVA, the variables are first classified as position variables, distance variables or mixed variables. Then, the distance variables are separated based on their interactions. However, in DVA, the individuals are randomly initialized; thus, the results of the analysis may be unstable. In addition, the interaction learning is not precise, and the corresponding non-separable groups cannot be correctly formed.

In GOPs, all variables should be optimized to obtain the single optimal (near-optimal) value. In MOPs, there is no need to optimize variables affecting the diversity; however, there is a need to assign different values to comprehensively approximate the PF. To converge to the Pareto optimal front (POF), the variables affecting the convergence should be optimized. Therefore, we allocate all diversity-related variables to one group, while we separate the convergence-related variables into several groups according to the interactions among them.

Many methods for grouping variables have emerged, such as fixed grouping [12], random grouping [17], the Delta method [18], dynamic grouping [19], differential grouping (DG) [20], global differential grouping (GDG) [21], gDG [14], an improved variant of DG (DG2) [22], etc. In DG-type methods, the interaction between each pair of variables is analyzed prior to optimization, which yields good grouping precision. gDG normalizes the differences and groups variables in the same connected subcomponents of the constructed graph together. This approach can correctly group all non-separable variables in the CEC′10 test suite [23].

Considering the accuracy of gDG, we extend it to MOLSOPs. In contrast to LSGOPs, in MOLSOPs, there are several objectives. For each objective, we learn the interactions among all variables. Then, we combine the interactions in all objectives together: if two variables interact with each other with respect to at least one objective, we consider there to be an interaction between them.

The contributions of this paper can be summarized as follows:

  • 1)

    We propose a new method called multiobjective graph-based differential grouping with shift (mogDG-shift) to decompose the variables in MOLSOPs.

  • 2)

    DVA and gDG are modified and combined to better recognize the properties of variables and the interactions among them and correctly form non-separable groups.

  • 3)

    We examine the grouping accuracy of mogDG-shift on MOLSOPs. It is verified that all non-separable variables can be recognized and that the corresponding groups can be correctly formed for LSMOP and DTLZ as well as almost all WFG instances. Compared with DVA, which is a state-of-the-art method, and the extended DG2, the precision of mogDG-shift is better, and less FEs are required.

  • 4)

    We combine mogDG-shift with MOEA/D and NSGA-II, and better performance is achieved.

The remainder of this paper is organized as follows. Section 2 provides the preliminary knowledge for this paper. The test suite used to validate the mogDG-shift method is introduced in Section 3. Section 4 presents the mogDG-shift method in detail. We perform grouping experiments in Section 5. Section 6 describes the optimization experiments and analyses. Additional experimentation is provided in Section 7. Finally, Section 8 concludes this paper.

Section snippets

Variable properties and separability

The variables in an MOP can be classified into three categories:

  • position variables, which affect diversity;

  • distance variables, which affect convergence; and

  • mixed variables, which simultaneously affect diversity and convergence.

For example, consider the following problem:f1(x)=x1cos(2.2πx2)+x2(x3+x4)f2(x)=1x1+sin(2.2πx2)+x2(x3+x4)s.t.xi[0,1],i=1,2,3,4

By varying the variables, we can obtain Fig. 1. When only x1 is varied, the generated solutions are non-dominated with respect to each other;

Large-scale multiobjective and many-objective test problems (LSMOPs)

Many multiobjective test suites have been developed, such as the Deb-Thiele-Laumanns-Zitzler (DTLZ) [24], Walking Fish Group (WFG) [15], and UF [25] suites. However, the test instances included in these test suites cannot be characterized as MOLSOPs. Recently, a collection of large-scale multiobjective and many-objective test problems (LSMOPs) [26] was proposed. These LSMOPs were constructed based on many concepts related to large-scale global optimization test suites (CEC′10 [23], CEC′13 [27],

Multiobjective graph-based differential grouping with shift (mogDG-shift)

In this section, we introduce the proposed multiobjective graph-based differential grouping with shift (mogDG-shift) technique. In mogDG-shift, there are three components: property analysis, interaction learning and graph-based grouping.

. Property Analysis.

Parameters in mogDG-shift

There are four parameters in mogDG-shift: the sampling number NPA, parameter γP, threshold ω and shift parameter γI.

The aim of variable property analysis is to classify variables as diversity-related (position variables and mixed variables) or convergence-related (distance variables) as introduced in Subsection 2.1 and detailed in Algorithm 1. For LSMOPs, the groundtruth is that the number of diversity-related variables is nObj − 1, all others are convergence-related variables, and there are no

Optimization experimentation on LSMOPs

In this section, we optimize the 2-/3-objective test instances in LSMOP using NSGA–II–DE and MOEA/D-DE [30] and then compare those using the same algorithms but combined with mogDG-shift and moDG2-shift, denoted as CCNSGA–II–DE and CCMOEA/D-DE, as well as CC2NSGA–II–DE and CC2MOEA/D-DE, respectively.

Additional experiments on other test suites

To further validate the proposed grouping method and improvement strategies, we also compare the aforementioned six algorithms with respect to additional multiobjective test suites: DTLZ [24] and WFG [15]. Corresponding to LSMOP, the objective numbers are 2 and 3, and the numbers of variables are 200 and 300, respectively. The same parameter settings described in the prior section are adopted here.

Conclusion

This paper proposes the mogDG-shift method to decompose the large number of variables in MOLSOPs. For the 2-/3-objective test instances in the LSMOP test suite, mogDG-shift is able to detect all non-separable variables and correctly form corresponding non-separable groups. To examine the contribution of mogDG-shift to the optimization performance, we combine mogDG-shift with NSGA–II–DE and MOEA/D-DE (denoted as CCNSGA–II–DE and CCMOEA/D-DE, respectively). The results of experiments show that

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grants Nos. 61976242 and 61379060, in part by the by the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University under Grant No. 2018002, in part by the Foundation of Key Laboratory of Machine Intelligence and Advanced Computing of the Ministry of Education under Grant No. MSC-201602A, and in part by the Special Program for Applied Research on

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