As an essential component in multi- and many-objective optimization, decision-making process either selects a subset of solutions from the whole Pareto front or guides the search toward a small part of the Pareto front during the evolutionary process. In recent years, for many-objective optimization problems (MaOPs), a number of evolutionary algorithms have been developed to search for Pareto optimal solutions. However, there is a lack of research works focusing on designing decision-making approaches. In order to overcome this deficiency, we propose a novel knee-based decision-making method to search for several solutions of interest (SOIs) from a large number of solutions on the Pareto front, each of which contains the best convergence performance at least within its neighborhood and can be identified as a global or local knee solution. The optimization performance achieved by all SOIs approximates the performance of the whole Pareto front as much as possible. Furthermore, in order to relieve the difficulties in the decision-making process on MaOPs, a new visualization approach is developed based on this proposed decision-making approach. It provides information about the shape and location of the Pareto front, the possible bulge, as well as the convergence degree and distribution of solutions. The experimental results on several benchmark functions demonstrate the superiority of the proposed design in the selection of SOIs and visualization of high-dimensional objective space.

中文翻译：

---

多目标优化中基于膝盖的决策和可视化

作为多目标和多目标优化的重要组成部分，决策过程要么从整个Pareto前沿中选择解决方案的子集，要么在进化过程中将搜索引导至Pareto前沿的一小部分。近年来，对于多目标优化问题（MaOP），已经开发了许多进化算法来搜索Pareto最优解。但是，缺乏专注于设计决策方法的研究工作。为了克服这一缺陷，我们提出了一种基于膝盖的新型决策方法，可以从Pareto前沿的大量解决方案中搜索几种感兴趣的解决方案（SOI），其中至少每个解决方案都包含最佳的收敛性能。它的邻域，可以识别为全局或局部膝盖解决方案。所有SOI所实现的优化性能都尽可能接近整个Pareto前沿的性能。此外，为了缓解MaOP决策过程中的困难，在此决策方法的基础上开发了一种新的可视化方法。它提供有关Pareto前沿的形状和位置，可能的凸出以及收敛度和解的分布的信息。在几个基准函数上的实验结果证明了所提出的设计在SOI的选择和高维目标空间可视化方面的优越性。为了减轻MaOPs决策过程中的困难，在此拟议决策方法的基础上开发了一种新的可视化方法。它提供有关Pareto前沿的形状和位置，可能的凸出以及收敛度和解的分布的信息。在几个基准函数上的实验结果证明了所提出的设计在SOI的选择和高维目标空间可视化方面的优越性。为了减轻MaOPs决策过程中的困难，在此拟议决策方法的基础上开发了一种新的可视化方法。它提供有关Pareto前沿的形状和位置，可能的凸出以及收敛度和解的分布的信息。在几个基准函数上的实验结果证明了所提出的设计在SOI的选择和高维目标空间可视化方面的优越性。