To tackle the challenge of maintaining population diversity in the decision space for multimodal multiohjective optimization prohlems (MMOPs), this paper introduces a novel mating selection strategy based on kernel K-means clustering. By leveraging the ability of kernel K-means to identify solution clusters within complex decision spaces, the proposed method enhances the exploration of promising parent solutions during the mating selection process. This method, combined with the environmental selection approach used in MMOEA/DC, led to the development of a novel algorithm called Multimodal Multiobjective Evolutionary Algorithm with Three Clustering mechanisms, termed as MMEATC. To evaluate the effectiveness of MMEATC, we compare MMEATC against five state-of-the-art multimodal multiobjective evolutionary algorithms: MMOEA/DC, MO_Ring_PSO_SCD, TriMOEA-TA&R, MMEA-WI, and CoMMEA. Extensive experimental results show that MMEATC achieves competitive performance, highlighting its potential as an effective approach to solving MMOPs.

Prominent achievements of multiagent reinforcement learning (MARL) have been recognized in the last few years, but effective cooperation among agents remains a challenge. Traditional methods neglect the modeling of action semantic relations in the learning process of joint action latent representations. In other words, the uncertain semantic relations might hinder the learning of sophisticated cooperative relationships among actions, which may lead to homogeneous behaviors across all agents and their limited exploration efficiency. Our aim is to learn the structure of the action semantic space to improve the cooperation‐aware representation for policy optimization of MARL. To achieve this, a scheme called graph learning of semantic relations (GLSR) is proposed, where action semantic embeddings and joint action representations are learned in a collaborative way. GLSR incorporates an action semantic encoder for capturing semantic relations in the action semantic space. By leveraging the cross‐attention mechanism with action semantic embeddings, GLSR prompts the action semantic relations to guide mining the cooperation‐aware joint action representations, implicitly facilitating agent cooperation in the joint policy space for more diverse behaviors of cooperative agents. The experimental results on challenging tasks demonstrate that GLSR attains state‐of‐the‐art outcomes and shows robust performance in multiagent cooperative tasks.

Most existing NAS-based multi-modal classification (MMC-NAS) methods are optimized using the classification accuracy.They can not simultaneously provide multiple models with diverse perferences such as model complex and classification performance for meeting different users' demands. Combining NAS-MMC with multi-objective optimization is a nature way for this issue. However, the challenge problem of this solution is the high computation cost. For multi-objective optimization, the computing bottleneck is pareto front search. Some higher-quality MMC models (namely core structures, CSs) consisting of high-quality features and fusion operators are easier to identify. We find that CSs have a close relation with the pareto front (PF), i.e., the individuals lying in PF contain the CSs. Based on the finding, we propose an efficient multi-objective neural architecture search for multi-modal classification by applying CSs to guide the PF search (CoMO-NAS). In conclusion, experimental results thoroughly demonstrate the effectiveness of our CoMO-NAS. Compared to state-of-the-art competitors on benchmark multi-modal tasks, we achieve comparable performance with lower model complexity in shorter search time.

Multimodal multiobjective optimization problems (MMOPs) possess multiple Pareto optimal sets (PSs) corresponding to the identical Pareto optimal front (PF). To handle MMOPs, we propose a bi-objective evolutionary algorithm (BOEA), which transforms an MMOP into a bi-objective optimization problem. This problem is constructed by the penalty boundary intersection technique and a diversity indicator to obtain multiple PSs. The first objective reflects the population convergence and factors in the population diversity in the objective space, while the other objective concentrates more on the population diversity in the decision space. Furthermore, an environmental selection strategy is designed to choose the offspring solutions, which consists of nondominated sorting based on the transformed optimization problem and hierarchical clustering for selecting promising solutions. Experiments on 34 MMOPs demonstrate that BOEA performs better than selected state-of-the-art representatives, including 22 MMOPs from CEC2019 and 12 imbalanced MMOPs. In addition, the effectiveness of BOEA is further validated by six feature selection problems in real-world applications.

An expensive multimodal optimization problem (EMMOP) is that the computation of the objective function is time consuming and it has multiple global optima. This article proposes a decomposition differential evolution (DE) based on the radial basis function (RBF) for EMMOPs, called D/REM. It mainly consists of two phases: the promising subregions detection (PSD) and the local search phase (LSP). In PSD, a population update strategy is designed and the mean-shift clustering is employed to predict the promising subregions of EMMOP. In LSP, a local RBF surrogate model is constructed for each promising subregion and each local RBF surrogate model tracks a global optimum of EMMOP. In this way, an EMMOP is decomposed into many expensive global optimization subproblems. To handle these subproblems, a popular DE variant, JADE, acts as the search engine to deal with these subproblems. A large number of numerical experiments unambiguously validate that D/REM can solve EMMOPs effectively and efficiently.

It is very difficult to locate multiple global optimal solutions (GOSs) of multimodal optimization problems (MMOPs). To deal with this issue, a penalty-based multimodal optimization differential evolution (DE), called PMODE, is developed in this article. In PMODE, a penalty strategy with a dynamic penalty radius is constructed to solve MMOPs. An elite selection mechanism is designed to identify and select elite solutions. The neighboring areas of these elite solutions are penalized. PMODE uses a popular DE variant-JADE as its search engine. The proposed PMODE is compared with several other state-of-the-art multimodal optimization algorithms on 20 MMOPs used in the IEEE CEC2013 special session. The experimental results show that PMODE performs better than other state-of-the-art methods.