IEEE Trans Cybern. 2018 Aug;48(8):2321-2334. doi: 10.1109/TCYB.2017.2737519. Epub 2017 Aug 18.
Multiobjective optimization problems with more than three objectives are commonly referred to as many-objective optimization problems (MaOPs). Development of algorithms to solve MaOPs has garnered significant research attention in recent years. "Decomposition" is a commonly adopted approach toward this aim, wherein the problem is divided into a set of simpler subproblems guided by a set of reference vectors. The reference vectors are often predefined and distributed uniformly in the objective space. Use of such uniform distribution of reference vectors has shown commendable performance on problems with "regular" Pareto optimal front (POF), i.e., those that are nondegenerate, smooth, continuous, and easily mapped by a unit simplex of reference vectors. However, the performance deteriorates for problems with "irregular" POF (i.e., which deviate from above properties), since a number of reference vectors may not have a solution on the POF along them. While adaptive approaches have been suggested in the literature that attempt to delete/insert reference directions conforming to the geometry of the evolving front, their performance may in turn be compromised for problems with regular POFs. This paper presents a generalized version of previously proposed decomposition-based evolutionary algorithm with adaptive reference vectors, intended toward achieving competitive performance for both types of problems. The proposed approach starts off with a set of uniform reference vectors and collects information about feasibility and nondominance of solutions that associate with the reference vectors over a learning period. Subsequently, new reference directions are inserted/deleted, while the original directions may assume an active or inactive role during the course of evolution. Numerical experiments are conducted over a wide range of problems with regular and irregular POFs with up to 15 objectives to demonstrate the competence of the proposed approach with the state-of-the-art methods.
具有三个以上目标的多目标优化问题通常被称为多目标优化问题（MaOPs）。近年来，开发用于解决 MaOPs 的算法引起了广泛的研究关注。“分解”是一种常用的方法，其中问题被划分为一组由一组参考向量引导的更简单的子问题。参考向量通常是预先定义的，并在目标空间中均匀分布。在具有“规则”Pareto 最优前沿（POF）的问题上，使用这种均匀分布的参考向量表现出色，即那些非退化、平滑、连续且易于由参考向量的单位单形映射的问题。然而，对于具有“不规则”POF（即偏离上述性质的问题）的性能会恶化，因为在 POF 上沿着它们可能没有一些参考向量的解。尽管文献中已经提出了自适应方法，试图根据不断演变的前沿的几何形状删除/插入参考方向，但对于具有规则 POF 的问题，它们的性能可能会受到影响。本文提出了一种基于分解的具有自适应参考向量的进化算法的广义版本，旨在为这两种类型的问题实现具有竞争力的性能。所提出的方法从一组均匀的参考向量开始，并在学习期间收集与参考向量相关的解的可行性和非支配性的信息。随后，插入/删除新的参考方向，而原始方向在进化过程中可能会扮演活跃或不活跃的角色。在具有规则和不规则 POF 的多达 15 个目标的广泛问题上进行了数值实验，以证明所提出的方法与最先进的方法的竞争力。
