Expensive constrained multiobjective optimization problems (ECMOPs) are prevalent in real-world scientific research and industrial applications. However, the complexity of feasible regions and the limitation on the number of available function evaluations often prevent most algorithms from achieving satisfactory results. To address these challenges, this article proposes an ensemble-based surrogate framework. Specifically, a global model and multiple local models are constructed as ensemble members to approximate each constraint function, aiming to improve the accuracy of landscape approximation for ECMOPs with complex feasible regions. Additionally, a novel vector-based constrained dominance principle is suggested to maintain the balance between objectives and constraints. By leveraging reference vectors, potential scenarios of the population during the evolutionary process are identified, and the customized selection strategy is devised for each scenario. These two techniques are integrated into a two-stage optimization framework, resulting in a surrogate-assisted evolutionary algorithm for solving ECMOPs. Through extensive experimental investigations, the proposed algorithm demonstrates significant superiority over seven other state-of-the-art peer algorithms on both benchmark test problems and real-world applications.

In expensive multitask optimization problems (ExMTOPs), multiple complex tasks must be optimized simultaneously under limited computational budgets. Existing approaches, often based on surrogate models, aim to approximate objective functions but struggle to generalize across heterogeneous tasks, depend on task-specific sampling, and require frequent retraining. To address these challenges, we propose the Multifactorial Evolutionary Algorithm–Single Step Generative Model (MFEA-SSG), a meta-learning-inspired framework that learns to generate high-quality solutions across tasks. Inspired by meta-learning, we treat each random shuffle of the decision variables as a unique pseudo-task, training the model on a distribution of these tasks to learn a task-agnostic prior about the structure of elite solutions. This process disrupts task-specific dependencies, allowing the model to learn transferable structures from recomposed samples. We then adopt a diffusion-based generative model to learn the distribution of optimal solutions, enabling knowledge transfer across tasks without directly approximating objective functions. To reduce inference cost, we introduce a student model distilled from the diffusion process. Unlike conventional diffusion models that denoise iteratively, the student generates solutions in a single forward pass, significantly reducing inference time. Comprehensive experiments on both general multitask benchmarks and a real-world protein mutation prediction scenario demonstrate that MFEA-SSG achieves high-quality solutions with fast convergence and low computational cost under limited evaluation budgets, outperforming state-of-the-art general and ExMTOPs algorithms.

In evolutionary multitasking, strategies such as crossover operators and skill factor assignment are critical for effective knowledge transfer. Existing improvements to crossover operators primarily focus on low-dimensional variable combinations, such as arithmetic crossover or partially mapped crossover, which are insufficient for modeling complex high-dimensional interactions. Moreover, static or semi-dynamic crossover strategies fail to adapt to the dynamic dependencies among tasks. In addition, current Multifactorial Evolutionary Algorithm frameworks often rely on fixed skill factor assignment strategies, lacking flexibility. To address these limitations, this paper proposes the Multifactorial Evolutionary AlgorithmResidual Learning (MFEA-RL) method based on residual learning. The method employs a Very Deep Super-Resolution (VDSR) model to generate high-dimensional residual representations of individuals, enhancing the modeling of complex relationships within dimensions. A ResNet-based mechanism dynamically assigns skill factors to improve task adaptability, while a random mapping mechanism efficiently performs crossover operations and mitigates the risk of negative transfer. Theoretical analysis and experimental results show that MFEA-RL outperforms state-of-the-art multitasking algorithms. It excels in both convergence and adaptability on standard evolutionary multitasking benchmarks, including CEC2017-MTSO and WCCI2020-MTSO. Additionally, its effectiveness is validated through a real-world application scenario.

Plenty of decision variable grouping based algorithms have shown satisfactory performance in solving high-dimensional optimization problems. However, most of them are tailored for inexpensive optimization problems. Extending variable grouping method to expensive optimization problems poses many challenges. One of the greatest challenges is that most grouping approaches require additional function evaluations (FEs) to discover interactions among decision variables, which is intolerable for expensive optimization problems as it incurs prohibitive computational costs. To address this issue, an adaptive variable grouping method is proposed in this paper, which can achieve relatively accurate grouping results without additional FE consumption. Specifically, variables are grouped based on the contrasts between well-converged solutions and poorly-converged solutions. Furthermore, the grouping scheme is adjusted dynamically during the optimization process to improve the grouping accuracy. Besides, an adaptive environmental selection based sampling strategy is suggested, which attempts to provide the currently required solutions for reevaluation according to the demands of different optimization stages. The proposed algorithm is compared with the other five state-of-the-art multiobjective optimization evolutionary algorithms on both benchmark problems and real-world problems. The experimental results demonstrate the promising performance and the superior computational efficiency of the proposed algorithm in tackling high-dimensional expensive multiobjective optimization problems.

With growing demand for privacy-preserving reinforcement learning (RL) applications, federated RL (FRL) has emerged as a potential solution. However, existing FRL methods struggle with multiple sources of heterogeneity, while lacking robust privacy guarantees. In this study, we propose DPA-FedRL, the dynamic privacy-aware FRL framework, to simultaneously mitigate both issues. First, we innovatively put forward the concept of "multiheterogeneity" and embed the environmental heterogeneity into agents' state representations. Next, to ensure privacy during model aggregation, we incorporate a differentially private mechanism in form of Gaussian noise and modify its global sensitivity, tailored to suit FRL's unique characteristics. Encouragingly, our approach dynamically allocates privacy budget based on heterogeneity levels, which strikes a balance between privacy and utility. From the theoretical perspective, we give rigorous convergence, privacy, and sensitivity guarantees for our proposed method. Through extensive experiments on diverse datasets, we demonstrate that DPA-FedRL surpasses state-of-the-art approaches (PPO-DP-SGD, PAvg, and QAvg) in some highly heterogeneous environments. Notably, our novel privacy attack simulations enable quantitative privacy assessment, validating that DPA-FedRL offers over $1.359\times $ stronger protection than baselines.

Large-scale sparse multiobjective optimization problems (SMOPs) exist widely in real-world applications, and solving them requires algorithms that can handle high-dimensional decision space while simultaneously discovering the sparse distribution of Pareto optimal solutions. However, it is difficult for most existing multiobjective evolutionary algorithms (MOEAs) to get satisfactory results. To address this problem, this article proposes a dynamic knowledge-guided coevolutionary algorithm, which employs a cooperative coevolutionary framework tailored for large-scale SMOPs. Specifically, variable selection is performed initially for the dimension reduction, and two populations are evolved in the original and reduced decision spaces, respectively. After offspring generation, variable replacement is performed to precisely identify the sparse distribution of Pareto optimal solutions. Furthermore, a dynamic score update mechanism is designed based on the discovered sparsity knowledge, which aims to adjust the direction of evolution dynamically. The superiority of the proposed algorithm is demonstrated by applying it to a variety of benchmark test instances and real-world test instances with the comparison of five other state-of-the-art MOEAs.

Social spider optimization (SSO) is a swarm algorithm designed for solving complex optimization problems. It is an effective approach for searching a global optimum by simulating the cooperative behavior of social-spiders. However, SSO takes too much computation time and shows premature convergence on some problems. In order to accelerate the computation speed and further enhance the search ability, a parallel SSO (PSSO) algorithm with emotional learning is proposed in this paper. First, we develop a parallel structure for the female and male individuals to update their positions, and each individual can be computed in parallel during the search process. Second, an emotional learning mechanism is used to increase swarm diversity which is helpful to improve the search performance. Furthermore, the convergence property and computational complexity of PSSO are discussed in detail. To test the effectiveness of the proposed algorithm, it is applied to solve data clustering problem. The experimental results demonstrate that the overall performance of PSSO is superior to six other clustering algorithms on several standard data sets. In the aspect of search performance, the results obtained by PSSO are better than the comparison algorithms in most used data sets. In the aspect of time performance, the computation time of PSSO is greatly reduced in the parallel computing environment. It is comparable with K-means which is the fastest among the comparison algorithms when the number of processors larger than and equals to 16.

The collective intelligent behaviors of insects or animal groups in nature have maintained the survival of the species for thousands of years. In this paper, a novel swarm intelligence algorithm called the social group entropy optimization (SGEO) algorithm is proposed for solving optimization tasks. The proposed algorithm is based on the social group model, the status optimization model, and the entropy model, which are the main contributions of this paper. First, the social group model and the feedback mechanism between Leaders and Followers are developed to reduce the probability of local optimum. Second, the status optimization model is described to reveal the changing rule about the population behavior states, to support the conversion between different social behaviors during evolution, to promote the algorithm to optimize quickly, and to avoid local optimization. Third, the entropy model is introduced to analyze the entropy of social groups, the change rule of difference entropy, and to set the information entropy as behavior's criterion of state optimization. In addition, the mathematical model of the SGEO is deduced from the group theory, matter dynamics, and the information entropy theory. The convergence and parallelism of it have been analyzed and verified theoretically. Moreover, to test the effectiveness of the SGEO, it is used to solve benchmark functions' problems that are commonly considered within the literature of evolutionary algorithms. Experimental results are compared with those of three other state-of-the-art algorithms. The superior performance of the SGEO validates its effectiveness and efficiency for the optimization problems, especially for the high-dimension problems.

Nature has always been a muse for those who dream in art or science. As it goes, optimization algorithms inspired by nature have been widely used to solve various scientific and engineering problems because of their intelligence and simplicity. As a novel nature‐inspired algorithm, the crystal energy optimizer (CEO) is proposed in this article. The proposed CEO is motivated by the following general observation on lake freezing in nature: the dynamics of crystals have possession of parallelism, openness, local interactivity, and self‐organization. It stimulates us to extend a crystal dynamic model in physics to a generalized crystal energy optimizer for traveling salesman problems, so as to exploit the advantages of crystal dynamic system and to realize the aforementioned purposes. The proposed CEO has these advantages: (1) it has the ability to perform large‐scale distributed parallel optimization; (2) it can converge and avoid local optimum; and (3) it is flexible and easy to adapt to a wide range of optimization problems.