@inproceedings {pub3835,
	title = {Adaptation of Reference Vectors for Evolutionary Many-objective Optimization of Problems with Irregular Pareto Fronts},
	author = {Qiqi Liu AND Yaochu Jin AND Martin Heiderich AND Tobias Rodemann},
	year = {2019},
	month = {June},
	abstract = {Most decomposition-based algorithms are designed based on the assumption that the Pareto Front (PF) has a regular geometrical structure, i.e, it is smooth, continuous, and well spread. Irregular problems are problems with discontinuous, degenerated, or inverted Pareto fronts, which just take up parts of the objective space. So when decomposition based algorithms are used to solve irregular problems, they do not perform well. The reason is that not all weight vectors or reference vectors work. In RVEA[1], every individual is associated with a nearest reference vector based on angle between reference vector and individuals. Therefore, it is very likely that several individuals are associated with one reference vector. As a result, some reference vectors are associated with no individuals. In this case, the distribution of the Pareto-optimal solutions can not be assured. To tackle problems with irregular PFs, a variant of RVEA, that is RVEA*[1], regenerates a set of reference vectors in addition to the predefined reference set. For invalid reference vectors, they are regenerated. As they are randomly generated in global space, a sufficient local solution density is not guaranteed. In this paper, we propose an improved algorithm, called iRVEA, to make it possible to tackle problems with irregular PFs. The main contributions of this paper can be summarized as follows. (a) When RVEA tackles problems with discontinuous PFs, it may only find a part of the discontinuous PF segments if solutions focus on some segments of PFs at the early stage of evolution (called {\textquoteright}global missing{\textquoteright} thereinafter). To solve this, we divide the whole objective space into several big sub regions and at least one solution is selected no matter whether it is non-dominated or not. (b) Besides the fixed reference vector set, we generate another non-fixed reference vector set. Reference vectors in non-fixed reference set are not fixed and can be adjusted as generation goes by to adapt to the distribution of solutions. Reference vectors in predefined and fixed
reference set are not adjusted during the course of evolution, as they are critical for keeping theuniformity of problems with regular PFs. In each generation, one invalid reference vector in the
non-fixed reference vector set is replaced by a solution in the population, which has the biggest angle to valid reference vectors in both fixed and non-fixed reference vector set.
(c) If a region is promising and some non-dominated individuals exist in this region while the valid reference vectors in this region are too few or even no reference vectors exist in this region,
it will lead to only a few individuals or no individuals in this region being selected (this phenomenon is called {\textquoteright}local missing{\textquoteright}). So only adopting APD selection in original RVEA is not
enough. In this paper, we use a Pareto dominance relationship for problems with up to eight objectives and use the SDR indicator[2] for problems with more than eight objectives to select
more solutions to alleviate the {\textquoteright}local missing{\textquoteright} phenomenon. The two main improvements of the iRVEA approach lie in the regeneration of invalid reference vectors
and environmental selection. For RVEA*, all reference vectors are likely to be adjusted in both the fixed and the non-fixed reference vector set and more than one invalid reference vectors is likely to be adjusted in every generation. In contrast, in our proposed algorithm, we just adjust at most one invalid reference vector in the non-fixed reference vector set and the valid reference vectors are kept unchanged. In environmental selection of RVEA*, solutions are selected only based on the APD selection criterion. However, in handling problems with irregular PFs, APD selection criterion is not enough to select enough solutions as the distribution of reference vectors may not be adjusted as promptly as the distribution of solutions in the population. To support APD selection, we add a second selection criterion to ensure that all promising solutions will be chosen. Moreover, in every generation, we use an archive to reserve solutions based on unary additive epsilon indicator [3] and the solutions in the archive are combined with solutions in the population for a second selection.},
	publisher = {IEEE},
	booktitle = {Proceedings of IEEE Conference on Evolutionary Computation (CEC)},
	pages = {1726-1733}
}