Elsevier

Information Sciences

Volume 512, February 2020, Pages 278-294
Information Sciences

Objective reduction for visualising many-objective solution sets

https://doi.org/10.1016/j.ins.2019.04.014Get rights and content

Abstract

Visualising a solution set is of high importance in many-objective optimisation. It can help algorithm designers understand the performance of search algorithms and decision makers select their preferred solution(s). In this paper, an objective reduction-based visualisation method (ORV) is proposed to view many-objective solution sets. ORV attempts to map a solution set from a high-dimensional objective space into a low-dimensional space while preserving the distribution and the Pareto dominance relation between solutions in the set. Specifically, ORV sequentially decomposes objective vectors which can be linearly represented by their positively correlated objective vectors until the expected number of preserved objective vectors is reached. ORV formulates the objective reduction as a solvable convex problem. Extensive experiments on both synthetic and real-world problems have verified the effectiveness of the proposed method.

Introduction

Many-objective optimisation problems (MaOPs) which involve more than three (conflicting) objectives to be optimised exist in many industrial and engineering applications [20]. Over the last decade, many-objective optimisation has attracted increasing interest in the evolutionary computation community [16], [20]. One inherent challenge in many-objective optimisation is that we cannot directly view solutions in such a four- or higher-dimensional objective space [28]. This brings difficulties for algorithm design, performance assessment, decision making, etc [21], [41].

In many-objective optimisation, the goal of visualising solutions is to enable researchers and practitioners to understand the given problem (e.g., the shape of its Pareto front) and the characteristics of the solutions (e.g., their Pareto dominance relation) [28]. In other words, a good visualisation can reveal the underlying structure of the problem, helps researchers and practitioners make a proper decision and illustrates the relation between solutions and also between objectives.

There are many multi-dimensional data visualisation methods, some of which have been applied to show many-objective solution sets, such as parallel coordinates [15], radar chart [18], [36], and heatmaps [29]. These methods directly plot the objective values of the solutions in a two-dimensional plane without any sophisticated transformations and can be easily extended to cases with higher dimensionality and to more solutions. However, in order to represent the conflicts between objectives, they require the objectives of interest to be positioned adjacent to each other. Also, the contour information of a given approximate Pareto-front is unavailable.

In recent years, dimension reduction-based visualisation techniques, which transform the input vectors into a lower-dimensional space, have attracted much interest from the evolutionary computation community. These techniques can help data analysts obtain new observations and insights through viewing the mapped data in the reduced space. To preserve the distribution of solutions, many dimension reduction methods have been applied, such as principal component analysis (PCA) [17], Sammon mapping [30], and neuroscale [26]. They do not care much about preserving the dominance relation between solutions and the conflicts between the objectives.

There exist several dimension reduction-based visualisation methods that consider the dominance relation between solutions. For instance, Köppen and Yoshida [19] used two different strategies to map the dominated and non-dominated solutions, respectively. He and Yen [13] proposed to map high-dimensional solutions into a 2D polar coordinate plot so that a large number of solutions can be viewed in a plane and the dominance relation between them can be approximately persevered. Tušar and Filipič [38] used the prosection approach to visualise four-dimensional solution sets, and it can preserve the dominance relation between solutions despite working only for problems with four objectives.

Two desirable properties in the many-objective solution set visualisation are preserving (1) the distribution of solutions and (2) the dominance relation between solutions. The distribution of solutions can be implied by the diversity and density of solutions and the contour information about the Pareto front. The dominance relation between solutions is a fundamental criterion of convergence, and considering it can help algorithm designer and decision maker select the optimal solutions.

In this paper, we propose an objective reduction-based visualisation method (ORV) for showing many-objective solution set. In ORV, we sequentially decompose objective vectors which can be linearly represented by their positively correlated objective vectors until the expected number of preserved objective vectors is reached. The main contributions and novelty of this work include the following:

  • An objective reduction-based method is proposed for viewing many-objective solution sets. The method is able to preserve the distribution of solutions and the dominance relation between solutions as far as possible.

  • The strategy of sequentially decomposing objective vectors makes the whole objective reduction process visible, which helps researchers and participators understand the conflicts between objectives through viewing the objective vector decomposition in each iteration.

  • It formulates the underlying objective vector representation as a convex optimisation problem, which can be solved efficiently. This formulation makes ORV capable of handling large-scale many-objective solution set.

  • Using ORV, algorithm designers can observe the evolutionary behaviour of their algorithms; decision makers can read the distribution of solutions, which help them in both quality evaluation and preference articulation processes.

The rest of this paper is organised as follows. Section 2 reviews the terminology and related work. Section 3 is devoted to the description of the new objective reduction method for displaying many-objective solution sets. Section 4 provides experimental results to illustrate the effectiveness of the proposed method. Finally, Section 5 concludes the paper.

Section snippets

Related work

Without loss of generality, a multi-objective optimisation problem (MOP) can be formulated as a minimisation problem and defined as follows:minF(x)=(f1(x),f2(x),,fM(x))Ts.t.xΩ,where ΩRn is the decision space, x=(x1,x2,,xn)T is a candidate solution, and F:ΩRM consists of M (conflicting) objective functions. The multi-objective optimisation problem with more than three conflicting objectives, i.e., M > 3, is referred to as a many-objective optimisation problem (MaOP).

Let a and b be two

Our proposed method

In this section, we present an objective reduction method to visualise a many-objective solution set. The basic idea of the proposed method is to decompose some objective vectors of a solution set into the remaining objective vectors in succession, thus forming a new set with lower dimensionality. In each iteration, we represent each objective vector by other objective vectors which are positively correlated with it. We then remove the objective vector that has the minimal representation error

Test problems

We evaluate the effectiveness of our method on both synthetic and real-world problems. The synthetic problems include the scalable DTLZ5(I, M) problem [6], a proposed test problem based on DTLZ7 [7], and the multi-line distance minimisation problem (ML-DMP) [22].

The DTLZ5(I, M) problem has the following three properties [6]: (1) the dimensionality of the Pareto front is I, where I is smaller or equal to the number of the problem’s objectives M, (2) the first MI+1 objectives are linear

Conclusion

Visualising a solution set of the many-objective optimisation problem is a challenging issue. Solutions lying in a high-dimensional space make them hard to be observed and perceived. This paper presented an objective reduction-based visualisation method, which tried to preserve the dominance and distribution relations between solutions during the objective reduction process. This allows users to visually know the behaviour of the solutions (e.g., their convergence and distribution shape) in

Acknowledgments

The authors would like to thank Prof. Robert M. Hierons and Dr. Sergio Segura for discussions on the SPL product selection problem. This work was supported by the National Natural Science Foundation of China under grants 61432012 and 61329302, the Engineering and Physical Sciences Research Council (EPSRC) of U.K. under grants EP/J017515/1 and EP/P005578/1, the Program for Guangdong Introducing Innovative and Entrepreneurial Teams (Grant no. 2017ZT07X386), Shenzhen Peacock Plan (Grant no.

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