多目标优化可视化方法比较分析

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"多目标优化中可视化方法的比较" 在多目标优化领域，可视化技术扮演着至关重要的角色。这是因为高维目标空间中的种群可视化能够为理解帕累托前沿提供关键洞察。多目标优化涉及在多个相互冲突的目标之间寻找平衡点，这通常是一个复杂的过程。可视化在这个过程中起到了两个主要作用：一是帮助开发更有效的多目标进化算法，通过在进化过程中的可视化，可以监控种群动态并指导算法改进；二是支持多准则决策制定，决策者可以直接观察每个解决方案的性能、目标之间的权衡以及近似前沿的分布，从而更容易地选择合适的解决方案。本文对现有的多目标优化可视化方法进行了详尽的总结，并将它们归类为五个不同的类别。这些类别可能包括： 1. **投影方法**：这类方法通过降维技术如主成分分析(PCA)或均匀散列映射(uMAP)将高维数据投影到低维空间，以便于观察和理解。 2. **分布图**：包括直方图、密度图等，用于展示目标值的分布情况，帮助识别帕累托前沿的形状和多样性。 3. **点云和散点图**：在二维或三维空间中显示个体点，可以直观地看到帕累托前沿的形状和种群的分布。 4. **彩色编码**：通过颜色的变化表示目标值或其他属性，增强视觉效果，帮助识别不同区域的解决方案特征。 5. **交互式界面**：允许用户动态探索和比较不同解决方案，提高决策的灵活性和参与度。为了评估这些可视化方法，作者设计了三个评价标准： 1. **信息保真度**：衡量方法在保留原始高维信息的同时降低维度的能力。 2. **可解释性**：评估方法是否能清晰地传达帕累托前沿的关键特性，使非专业用户也能理解。 3. **实用性**：考虑方法在实际应用中的效率和用户友好性，包括计算复杂度和交互性。通过对各种方法进行比较和评估，研究人员和工程师可以根据具体需求选择最适合的可视化策略。此外，该论文还讨论了现有方法的局限性和未来研究的方向，比如如何更好地处理高维数据、如何提高对拥挤帕累托前沿的可视化效果，以及如何结合人工智能和机器学习技术提升可视化质量。多目标优化中的可视化方法是理解和解决复杂优化问题的关键工具，通过比较和分析，我们可以更好地利用这些方法来优化算法设计和决策过程。这篇研究论文为这个领域的研究提供了宝贵的参考和指导。

Comparison of Visualization Approaches in Many-

objective Optimization

Zhenan He

College of Computer Science

Sichuan University

Chengdu, Sichuan, CHINA

zhenan@scu.edu.cn

Gary G. Yen

School of Electrical and Computer Engineering

Oklahoma State University

Stillwater, Oklahoma, USA

gyen@okstate.edu

Abstract—In many-objective optimization, visualization of

population in the high-dimensional objective space provides a

critical understanding of the Pareto front. First, visualization

throughout the evolutionary process can be exploited in

developing effective many-objective evolutionary algorithms.

Furthermore, visualization is a crucial component of multi-

criteria decision making. By directly observing the performance

of each solution, the trade-off between objectives, and

distribution of the approximate front, the decision maker can

easily decide which solution should be chosen from. In this

paper, we make a detailed summary for existing visualization

approaches and group them into five different categories. Then,

three evaluation criteria for visualization approaches are

designed, according to which, five state-of-the-arts are

compared under the created data sets. Experimental results

show that all approaches can satisfy each criterion to some

degree but no one can fully achieve all of these criteria. There is

a need to develop the new approach emphasis on fully satisfy all

criteria simultaneously. Then, based on the comparison results,

two future research directions for visualization approach are

proposed.

Keywords—many-objective optimization; visualization;

decision making;

I. INTRODUCTION

Many-objective Optimization Problems (MaOPs) involve

more than three conflicting objectives to be optimized and

widely exist in real-world applications [1-3]. On the other

hand, Multi-objective Optimization Problems (MOPs) contain

only two or three objectives, the low dimensional objective

space of which can be easily visualized by scatter plot with

each axis represents one objective, providing the information

of location, distribution, and shape for the approximate Pareto

front. However, in MaOPs, due to the curse of dimensionality,

the scatter plot is no longer an available tool in visualization

of a high dimensional objective space [4].

For many-objective optimization, it is strongly required to

visualize population in a high dimensional objective space,

not only for the development of the effective Many-Objective

Evolutionary Algorithms (MaOEAs), but also for

performance evaluation of MaOEAs and decision making.

Therefore, a high quality visualization tool must provide three

types of information in the high dimensional space. First, it

should provide accurate shape, location, and range of the

approximate Pareto front [4]. Second, from it, decision makers

can observe trade-off between objectives, monitor the

evolution progress, assess the quality of the approximate front,

and select their preferred solutions if desired [5]. Third, this

visualization tool itself should be scalable to any dimensions,

handle a large number of individuals on the approximate front,

and simultaneously visualize multiple fronts for the purpose

of visual comparison. Moreover, the resulted visualization

plot should be robust and insensitive to the addition or

removal of an individual.

In the literature, in order to view high dimensional data,

many visualization approaches have been proposed. First,

Buddle chart [6] extends the scatter plot from visualization of

three dimensional space to five dimensional space using size

and color to represent the fourth and fifth dimensions,

respectively. However, it cannot be extended further more. On

the other hand, in a M-dimension objective space, where M is

any integer number, Parallel Coordinates [7] and Heatmaps

[8] represent a M-dimension solution on a two-dimension

parallel coordinate system with M parallel axes representing

M objectives [5]. Then, the original objective values of

solutions can be retained. Borrowing the idea from Physics,

RadViz [9] transforms each solution’s objective value in high

dimensional space to the distance between the solutions and

that objective in the mapped two dimensional space. Sammon

Mapping [10] and Multi-Dimensional Scaling [11] construct

a mapping from a high dimensional objective space into a two-

dimension space by preserving local relations between each

pair of solutions in a high dimensional space. Finally, Self-

Organizing Map [12] and Neuroscale [13] apply surrogate

model to accomplish the mapping from high dimensional

space to lower dimension.

In this paper, we present a summary of existing

visualization approaches in literature and group them into five

different categories. Then, three evaluation criteria are

designed, each of which focuses on how the visualization

method reflects the information of the original front, how the

visualization result is helpful to decision making, and the

characteristics of the visualization approach. According to

these criteria, four state-of-the-arts are compared under four

created data sets. Finally, based on the comparison results, two

future research directions for visualization approach are

proposed. First, ensemble a group of visualization approaches

This work was supported in part by the National Nature Science Foundation

of China under grant 61602328, and by the Sichuan University under grant

2082204194216.

357

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