Abstract
In the real world, multi-objective optimization problems (MOPs) are very common and often involve multiple conflicting objectives. Consequently, no solutions can simultaneously satisfy all the objectives but a trade-off solutions will be obtained. The conventional multiobjective evolutionary algorithms (MOEAs) are dedicated to finding a solution set with a good balance between the convergence and diversity to represent the Pareto optimal front (PoF). However, in practice, the decision-maker (DM) may be only interested in some parts of the PoF. Accordingly, the past decades of years have witnessed the development of the preference-driven MOEAs, seeking several solutions or regions of the PoF of the MOPs to satisfy the preference from the DM. Notably, the DMs may face a great challenge in the articulation of explicit preference, when they have insufficient a priori knowledge of the problems. Therefore, the search of natural solutions of interest such as the knee points has become a new line of research in recent years. Nevertheless, little work has been reported focusing on designing multi-objective problems whose Pareto front contains complex knee regions. Likewise, few performance indicators dedicated to evaluating an algorithm's ability of accurately locating all knee points in high-dimensional objective space have been suggested. Additionally, the a posteriori knee identification methods implicitly assume that the given solutions are well distributed over the whole Pareto optimal front (PoF) and able to provide sufficient information for identifying the knee solutions. However, this assumption may fail in practice, in particular when the number of objectives is very large or when the shape of the PoF is complex. Furthermore, most a priori methods mainly search knee regions in low-dimensional objective spaces and fail to achieve good performance in locating the knee regions in high-dimensional objective space. Accordingly, this thesis aims to fill the above gaps. To begin with, we proposed a set of multi-objective optimization test problems which Pareto front consists of complex knee regions, aiming to assess the capability of evolutionary algorithms to accurately identify all knee points. Various features related to knee points have been taken into account in designing the test problems, including symmetry, differentiability, degeneration. These features are also combined with other challenges in solving optimization problems, such as multimodality, linkage between decision variables, non-uniformity and scalability of the Pareto front. The proposed test problems are scalable to both decision and objective spaces. Accordingly, new performance indicators are suggested for evaluating the capability of optimization algorithms in locating the knee points. The proposed test problems together with the performance indicators offer a new means to develop and assess preference-based evolutionary algorithms for solving multiand many-objective optimization problems. After that, an a posteriori MOEA has been proposed to alleviate the concern from the assumption. The basic idea is to augment the given solution set by generating solutions near the promising knee regions, thereby improving the performance of knee point identification. In the method, we first transform the PoF into a multimodal auxiliary function, whose minimums correspond to the knee points of the PoF. Then, a surrogate model is built to approximate the auxiliary function and a variant of differential evolution is employed to search the basins of the approximated auxiliary functions, so that additional solutions in the detected basins can be generated. After that, these new solutions in the objective space are mapped to the decision space with the help of an inverse model and are evaluated by the original objective functions. Finally they are added to update the given solution set. Besides, a new method is introduced to search the knee candidates in terms of the above augmentation strategy. Accordingly, the performance of the proposed and other knee identification methods will be greatly improved, and the concerns of the assumption will be eased to much extent. Additionally, an a priori MOEA using two localized dominance relationships has been proposed for the search of knee regions in high-dimensional objective space. In the environmental selection, the a-dominance is applied to each subpopulation partitioned by a set of predefined reference vectors, thereby guiding the search towards different potential knee regions while removing possible dominance resistant solutions. A knee-oriented dominance measure making use of the extreme points is then proposed to detect knee solutions in convex knee regions and discard solutions in concave knee regions. Without the misleading from the discarded solutions, the search process can be guided to the potential knee regions and the knee candidates can be found in high-dimensional objective space. Consequently, the knee candidates in high-dimensional objective space will be found by the proposed method. Finally, we also conduct investigations of the proposed methods on a real application (hybrid electrical vehicle controller design problem with seven objectives). This study provides an insight into dealing with MOPs or MaOPs when the DM cannot specify explicit preference, and hopefully contributes the EMO, especially preference-driven EMO to much extent.