Feature selection is a complex optimization problem with important real-world applications. Normally, its main target is to reduce the dimensionality of the dataset and increase the effectiveness of the classification. Owing to the population-inspired characteristics, different evolutionary algorithms (EAs) have been proposed to solve feature selection problems over the past decades. However, the majority of them only consider single objective optimization while many real-world problems have multiple objectives, which creates a genuine demand for designing more suitable and effective EAs to handle multi-objective feature selection. A multi-objective feature selection problem usually consists of two objectives: one is to minimize the number of selected features and the other is to minimize the error of classification. In this paper, we propose a duplication analysis based evolutionary algorithm (termed DAEA) for bi-objective feature selection in classification. In the proposed algorithm, we make improvements on the basic dominance-based EA framework in three aspects: first, the reproduction process is modified to improve the quality of offspring; second, a duplication analysis method is proposed to filter out the redundant solutions; third, a diversity based selection method is adopted to further select the reserved solutions. In the experiments, we have compared the proposed algorithm with five state-of-the-art multi-objective evolutionary algorithms (MOEAs) and tested them on 20 classification datasets, using two widely-used performance metrics. According to the empirical results, DAEA performs the best on most datasets, indicating that DAEA not only gains outstanding optimization performance but also obtains good classification and generalisation results.

Recently, numerous multiobjective evolutionary algorithms (MOEAs) have been proposed to solve the multiobjective optimization problems (MOPs). One of the most widely studied MOEAs is that based on decomposition (MOEA/D), which decomposes an MOP into a series of scalar optimization subproblems, via a set of uniformly distributed weight vectors. MOEA/D shows excellent performance on most mild MOPs, but may face difficulties on ill MOPs, with complex Pareto fronts, which are pointed, long tailed, disconnected, or degenerate. That is because the weight vectors used in decomposition are all preset and invariant. To overcome it, a new MOEA based on hierarchical decomposition (MOEA/HD) is proposed in this paper. In MOEA/HD, subproblems are layered into different hierarchies, and the search directions of lower-hierarchy subproblems are adaptively adjusted, according to the higher-hierarchy search results. In the experiments, MOEA/HD is compared with four state-of-the-art MOEAs, in terms of two widely used performance metrics. According to the empirical results, MOEA/HD shows promising performance on all the test problems.

The existing multiobjective evolutionary algorithms (EAs) based on nondominated sorting may encounter serious difficulties in tackling many-objective optimization problems (MaOPs), because the number of nondominated solutions increases exponentially with the number of objectives, leading to a severe loss of selection pressure. To address this problem, some existing many-objective EAs (MaOEAs) adopt Euclidean or Manhattan distance to estimate the convergence of each solution during the environmental selection process. Nevertheless, either Euclidean or Manhattan distance is a special case of Minkowski distance with the order P = 2 or P = 1, respectively. Thus, it is natural to adopt Minkowski distance for convergence estimation, in order to cover various types of Pareto fronts (PFs) with different concavity-convexity degrees. In this paper, a Minkowski distance-based EA is proposed to solve MaOPs. In the proposed algorithm, first, the concavity-convexity degree of the approximate PF, denoted by the value of P, is dynamically estimated. Subsequently, the Minkowski distance of order P is used to estimate the convergence of each solution. Finally, the optimal solutions are selected by a comprehensive method, based on both convergence and diversity. In the experiments, the proposed algorithm is compared with five state-of-the-art MaOEAs on some widely used benchmark problems. Moreover, the modified versions for two compared algorithms, integrated with the proposed P-estimation method and the Minkowski distance, are also designed and analyzed. Empirical results show that the proposed algorithm is very competitive against other MaOEAs for solving MaOPs, and two modified compared algorithms are generally more effective than their predecessors.

Over the past two decades, numerous multi- and many-objective evolutionary algorithms (MOEAs and MaOEAs) have been proposed to solve the multi- and many-objective optimization problems (MOPs and MaOPs), respectively. It is known that the difficulty of maintaining the convergence and diversity performances rapidly grows as the number of objectives increases. This phenomenon is especially evident for the Pareto-dominance-based EAs, because the nondominated sorting often fails to provide enough convergent pressure toward the Pareto front (PF). Therefore, many researchers came up with some non-Pareto-dominance-based EAs, which are based on indicator, decomposition, and so on. In this article, we propose a polar-metric ( p-metric)-based EA (PMEA) for tackling both MOPs and MaOPs. p-metric is a recently proposed performance indicator which adopts a set of uniformly distributed direction vectors. In PMEA, we use a two-phase selection which combines both nondominated sorting and p-metric. Moreover, a modification is proposed to adjust the direction vectors of p-metric dynamically. In the experiments, PMEA is compared with six state-of-the-art EAs in total and is measured by three performance metrics, including p-metric. According to the empirical results, PMEA shows promising performances on most of the test problems, involving both MOPs and MaOPs.

High dimensionality often challenges the efficiency and accuracy of a classifier, while evolutionary feature selection is an effective method for data preprocessing and dimensionality reduction. However, with the exponential expansion of search space along with the increase of features, traditional evolutionary feature selection methods could still find it difficult to search for optimal or near optimal solutions in the large-scale search space. To overcome the above issue, in this paper, we propose a bi-search evolutionary algorithm (termed BSEA) for tackling high-dimensional feature selection in classification, with two contradictory optimizing objectives (i.e., minimizing both selected features and classification errors). In BSEA, a bi-search evolutionary mode combining the forward and backward searching tasks is adopted to enhance the search ability in the large-scale search space; in addition, an adaptive feature analysis mechanism is also designed to the explore promising features for efficiently reproducing more diverse offspring. In the experiments, BSEA is comprehensively compared with 9 most recent or classic state-of-the-art MOEAs on a series of 11 high-dimensional datasets with no less than 2000 features. The empirical results suggest that BSEA generally performs the best on most of the datasets in terms of all performance metrics, along with high computational efficiency, while each of its essential components can take positive effect on boosting the search ability and together make the best contribution.

In this paper, we propose a sampling approach of reference points used for performance metrics of multi-objective evolutionary algorithms. Traditional reference point sampling methods, such as the Das and Dennis method, usually sample the reference points via a set of uniformly distributed weight vectors generated on an ideal hyper-plane in objective space, which however often ignore the geometric shape of a specific Pareto front. Therefore, we propose a novel reference point sampling approach by taking the specific shape of the Pareto optimal front to be tackled into account for measuring the performance of multi-objective evolutionary algorithms. The performance of the proposed reference point sampling method against the other two state-of-the-art sampling methods is tested on six test instances in various conditions, which clearly demonstrate the effectiveness and superiority of the proposed sampling method.