Over the past few years, a variety of surrogate-assisted evolutionary algorithms have emerged, aiming to tackle expensive multi-objective optimization problems. On the other hand, as the problem's dimensionality rises, the sample size needed for training surrogate models also increases significantly, making this approach impractical for large-scale scenarios. To overcome this challenge, we propose a large-scale expensive multi-objective optimization algorithm leverages collaboration among multiple subproblems. Specifically, we employ random feature selection with multiple times to partition the original decision space into multiple subproblems. For each objective, a surrogate model is constructed based on the historical data of each subproblem. After searching for the surrogate models of each subproblem, the population of original problem is updated. Additionally, to accelerate optimization, we select m individuals with the best performance for each objective, as well as the one nearest to the ideal point, for true evaluations. To illustrate the effectiveness of our approach, comparative experiments have been carried out using the DTLZ and ZDT benchmark problems, which involve up to 200 decision variables. The experimental outcomes indicate that the proposed technique delivers strong performance in addressing large-scale, expensive multi-objective optimization tasks.

Machine learning models based on artificial neural networks have been widely adopted to support diverse complex applications. However, the training of such models heavily relies on large-scale datasets, which may raise privacy concerns. Taking the Internet of Things (IoT) as an example, distributed edge devices collect data, while central servers aggregate this data for model training and in-depth analysis. Unlike frameworks like federated learning, local model training becomes infeasible due to constrained edge device capabilities, model confidentiality requirements, or the separation of complete data features across multiple devices. This scenario may render traditional federated learning-based privacy-preserving frameworks ineffective. To address these limitations, we propose an efficient privacy-preserving model training protocol that demonstrates practical advantages over fully homomorphic encryption and functional encryption methods. Our protocol leverages additively homomorphic encryption combined with the Chinese Remainder Theorem to design secure matrix-vector computations, minimizing encryption/decryption operations and reducing computational overhead on edge devices while preserving training efficiency and model accuracy. To further optimize performance, we further propose a secondary protocol that introduces optimization strategies to enhance efficiency and lighten edge nodes’ computational burdens. Security analysis and rigorous correctness proofs are provided, and performance evaluations and experimental validation confirm both protocols’ practical feasibility and effectiveness.