In recent years, multimodal multi-objective optimization problems (MMOPs) have become a prominent research focus in the field of computational evolution, increasing the demands on the performance of multimodal multi-objective evolutionary algorithms (MMOEAs). To evaluate the merits of MMOEA, it is usually necessary to satisfy the following three key criteria: (1) good convergence, (2) the ability to find more equivalent Pareto solutions (PSs), and (3) a uniform population distribution in the decision space and objective space. However, most current algorithms fail to satisfy all the above criteria simultaneously when facing challenges such as search tasks of varying difficulty and uneven allocation of computational resources. To address these challenges, this paper proposes an enhanced goal direction and remodeling of population distributions multimodal multi-objective evolutionary algorithm (MMOEA-EGR). The algorithm dynamically selects evolutionary stages through reinforcement learning, flexibly guiding the population to evolve under the guidance of the objectives of each stage, thus promoting efficient collaboration among the stages in the whole optimization process. Meanwhile, the algorithm adopts a remodeling population distribution strategy to enhance the evolutionary efficiency while optimizing the diversity of the decision space. Experimental results show that MMOEA-EGR outperforms several mainstream multimodal multi-objective evolutionary algorithms on several MMOPs standard test sets.

Tumor classification based on microarray gene expression data is easy to fall into overfitting because such data are composed of many irrelevant, redundant, and noisy genes. Traditional gene selection methods cannot achieve satisfactory classification results. In this study, we propose a novel multi-target hybrid gene selection method named RMOGA (ReliefF Multi-Objective Genetic Algorithm), which aims to select a few genes and obtain good tumor recognition accuracy. RMOGA consists of two phases. Firstly, ReliefF is used to select the top 5% subset of genes from the original datasets. Secondly, a multi-objective genetic algorithm searches for the optimal gene subset from the gene subset obtained by the ReliefF method. To verify the validity of RMOGA, we conducted extensive experiments on 11 available microarray datasets and compared the proposed method with other previous methods. Two classical classifiers including Naive Bayes and Support Vector Machine were used to measure the classification performance of all comparison methods. Experimental results show that the RMOGA algorithm can yield significantly better results than previous state-of-the-art methods in terms of classification accuracy and the number of selected genes.

Formalization of rough sets is a key issue in rough set theory. When rough sets are formalized by propositional logic, predicate logic, or modal propositional logic, it easily suffers from some problems. For instance, an incomplete system is obtained. The concepts of “ p r e c i s e” or “ r o u g h” of rough sets cannot be described. To tackle these issues, a new noncontingency axiomatic system is proposed for formalizing rough sets in this paper. First, a new concise accessibility relation is defined for the axiomatic system; then, two simpler axiom schemas of the axiomatic system are designed to replace the axiom schema K. This is helpful to prove the soundness and completeness theorems for the axiomatic system. Finally, rough sets can be perfectly formalized by our proposed axiomatic system. Theoretical analysis proves that a complete formal system is achieved. In addition, the concepts of “ p r e c i s e” or “ r o u g h” of rough sets can be described without the help of semantics functions of metalanguage.