Time series forecasting faces a fundamental challenge: the uneven distribution of predictive importance in time series data, where some specific time points and feature combinations carry disproportionately predictive power. As a result, uniform processing methods that treat all data alike inevitably fall short of optimal performance. To address this problem, we propose FeTS, a feature-aware framework that comprehensively learns temporal features through two key components: (i) Adaptive Feature Extraction (AdaFE), which dynamically discovers the most important features within each temporal patch and extracts them on the fly, yielding sharper and more focused local representations; and (ii) Dual-Scale Feed-Forward Network (DSFFN), which strategically integrates fine-grained local features with global long-term dependencies to achieve richer dual-scale representation learning. Extensive experiments on eight benchmark datasets demonstrate that FeTS achieves state-of-the-art performance in time series forecasting tasks, offering a novel solution to the challenge of uneven predictive importance in forecasting.

Recently, machine learning-embedded large-scale multiobjective evolutionary algorithms (LMOEAs) have shown great promise in solving large-scale multiobjective optimization problems (LMOPs). However, the fast convergence of the population to the true Pareto-optimal front (POF) and even distribution of the obtained Pareto-optimal solutions (POSs) on the POF are not adequately considered when tackling an LMOP. Besides, existing LMOEAs typically pair solutions with a matching rule and employ a network to learn the evolution pattern among the obtained solution pairs. It is difficult to learn various evolution patterns through a simple network, which hinders the collaboration of different patterns for enhancing the search capability. Facing such difficulties, this article proposes an LMOEA with multipattern learning and collaboration (LMOEA-MLC), where a single-hidden-layer multioutput network (SMN) is established to learn inductive and hybrid evolution patterns. Specifically, two inductive ones can be learned with the solution pairs built by two matching rules toward fast convergence and even distribution, respectively. Moreover, the solution pairs considering the fusion of the two inductive ones are collected, enabling SMN to learn a hybrid one and thus making a tradeoff between fast convergence and even distribution. Besides, the learned evolution patterns collaborate to enhance the search capability due to the distinct patterns. To enhance learning speed, SMN’s parameters are updated by an incremental random vector functional link (IRVFL). In our experiments, comprehensive comparisons with eight state-of-the-art LMOEAs demonstrate the significant performance improvement of LMOEA-MLC in handling LMOPs.

Training graph neural networks (GNNs) for graph representation has received increasing concerns due to its outstanding performance in the link prediction and node classification tasks, but it incurs much time and storage for tackling large-scale graphs. To alleviate this issue, graph condensation has been emerged to condense the large graph into a small but highly-informative graph, while achieving comparable performance of GNNs trained on the small graph and large graph. However, existing works mainly focus on the gradient or distribution matching under GNN training trajectories to condense simple link structures, while overlooking the structure matching for condensing signed graph that exists conflict links and structural balance among nodes. To bridge this gap, we propose a novel Structure Balance and Gradient Matching-Based Signed Graph Condensation (SGSGC) method for condensing signed graph with node attributes, conflict links and structural balance into informative smaller ones. Specifically, we first propose a structure-balanced matching to match the structural balance between the original and condensed signed graph, and then combine it with the gradient matching to condense signed graph for the link sign prediction task, while preserving both conflicting link structures and node attributes. Moreover, we use the feature smoothing and the graph sparsification technique to improve the robustness for the GNN training, respectively. Finally, a bi-level optimization technique is proposed to simultaneously find the optimal node attributes and conflict structure of the condensed graph. Experiments on six datasets demonstrate that SGSGC achieves excellent performance. On Epinions, 94% test accuracy of training on the original signed graph, while reducing their graph size by 99.95% - 99.99%, and there exist 2.24% – 6.26% accuracy improvements for link sign prediction compared to the state-of-the-arts.

This paper proposes a multiobjective many-tasking evolutionary algorithm with enhanced search direction-based knowledge transfer (MMaTEA-ESD). Specifically, the search directions for all tasks are first dynamically computed based on the populations obtained during the evolutionary search process. Subsequently, the search direction of the target task is divided into multiple subsearch directions. After that, the most similar subsearch directions from candidate source tasks are identified by measuring their similarity to those of the target task. Finally, the selected subsearch directions are combined to form the enhanced search direction for knowledge transfer, effectively mitigating negative transfer. Numerical results on a widely used multiobjective many-tasking benchmark test suite demonstrate the competitive performance of MMaTEA-ESD over some state-of-the-art algorithms.

The existing constrained multiobjective evolutionary algorithms (CMOEAs) still have great room for improvement in balancing populations convergence, diversity and feasibility on complex constrained multiobjective optimization problems (CMOPs). Besides, their effectiveness deteriorates dramatically when facing the CMOPs with scaling-up objective space or search space. We are thus motivated to design a learning-aided CMOEA with promising problem-solving ability and scalability for various CMOPs. In the proposed solver, two learning models are respectively trained online on constrained-ignored task and feasibility-first task, which are then used to learn the two improvement-based vectors for enhancing the search by differential evolution. In addition, the union population of parent and child solutions is divided into multiple subsets with a hierarchical clustering based on cosine similarity. A comprehensive indicator, considering objective-based performance and constraint violation degree of a solution, is developed to select the representative solution from each cluster. The effectiveness of the proposed optimizer is verified by solving the CMOPs with various irregular Pareto fronts, the number of objectives ranging from 2 to 15, and the dimensionality of search space scaling up to 1000.

Surrogate-assisted evolutionary algorithms (SAEAs) have shown promising performance in tackling expensive multiobjective optimization problems (EMOPs). However, existing SAEAs solve EMOPs separately, which ignore their optimization experiences earned before. Inspired by multitasking optimization paradigm for multitasking multiobjective optimization problems (MTMOPs), this article designs the first SAEA for tackling expensive MTMOPs (EMTMOPs) with adaptive knowledge transfer. First, a competitive surrogate selection is proposed to improve the generalization ability of approximating various EMOP tasks, where two types of surrogate models are trained and then compete for use to replace real expensive evaluations. Then, an adaptive solution selection is designed, which identifies promising transfer solutions to accelerate the solving of target task and selects promising infill solutions for real expensive evaluations to refine the surrogate models. The performance of our algorithm is validated on three commonly used benchmark suites and some real-world EMTMOPs. The experiments validate our superiority over several state-of-the-art SAEAs on most test cases.

In recent years, dynamic multiobjective evolutionary algorithms (DMOEAs) using the prediction strategy have shown promising performance for solving dynamic multiobjective optimization problems (DMOPs), as they can predict environmental changing trends in advance. However, most of them follow a regular change pattern and thus their performance is compromised when solving DMOPs with irregular change patterns (e.g., nonlinear correlations). To alleviate this challenge, this article proposes a DMOEA with a learnable prediction for tackling DMOPs. Specifically, a neural network is designed to effectively capture diverse change patterns of the environment. Based on the change patterns learned, a directional improvement prediction (DIP) is developed to guide the evolutionary search toward promising directions in the decision space. In this way, a superior initial population with good convergence and diversity is predicted by DIP, which can be more effective for solving various DMOPs. Comprehensive empirical studies show that the proposed DIP is effective and the proposed algorithm has some advantages over five competitive DMOEAs when solving three commonly used benchmarks and one real-world problem.

To enhance the generalization of multi-objective feature selection (MOFS) in classification, this paper proposes an evolutionary multitasking algorithm, diverging from previous approaches that exclusively target selected features. The algorithm integrates information from both selected and unselected features, introducing a novel objective to minimize the accuracy of unselected features. This objective, combined with the goal of minimizing classification errors for selected features, forms an auxiliary MOFS task. The paper presents a dual-population evolutionary multitasking framework that synergizes the main MOFS task with the auxiliary task. A knowledge transfer mechanism, based on accuracy preferences, seamlessly shares insights from the auxiliary to the main task, aiming to identify improved Pareto feature subsets. Empirical results demonstrate the superior performance of several state-of-the-art multi-objective algorithms within this framework, highlighting significant improvements across diverse datasets.

Surrogate-assisted evolutionary algorithms (SAEAs) have demonstrated promising optimization performance in addressing expensive dynamic optimization problems or expensive multimodal optimization problems. However, none of existing SAEAs are designed specifically for tackling expensive dynamic multimodal optimization problems (EDMMOPs). Therefore, in this paper, a first SAEA for tackling EDMMOPs is proposed. First, a nearest density clustering is designed to divide the population into a number of subpopulations, enhancing the diversity of the population. Then, a surrogate-assisted evolutionary optimizer is developed to construct surrogate models for each subpopulation and evolve all solutions in subpopulations by means of the built surrogate models, accelerating the population's converge towards several optimal solutions rapidly. Finally, a transfer learning-based prediction is devised to generate initial samples for next environment by leveraging the stored training samples in the previous environments. To assess the performance of our proposed algorithm, a set of complex benchmark problems is adopted, and the experimental results confirm its superior performance over several competitive algorithms on most test cases.

In cases of frequent problem-solving of multiobjective optimization tasks from a domain due to changing conditions or problem features, a growing number of individual tasks will be solved and stored in a database, providing an opportunity for a target task at hand to achieve better optimization performance through knowledge transfer from the previously-solved tasks, which is also known as sequential transfer optimization. Despite a variety of transfer algorithms that have been developed over the years, the research on the design of benchmark problems for evaluating such algorithms received far less attention. Oftentimes, the source and target tasks in existing test problems are manually assembled or extended from specific practical problems, limiting their ability to represent the diverse yet complex source-target similarity relationships in real-world problems. In light of this, we propose design methods to generate multiobjective sequential transfer optimization problems (MSTOPs) systematically in this work, wherein the Pareto manifolds of individual tasks and the manifold-based similarity between the tasks can be customized with ease, enabling a broad spectrum of representation of the diverse similarity relationships between the source-target Pareto manifolds of MSTOPs. Lastly, a benchmark suite with 12 test problems is developed using the proposed methods, which would serve as an arena for electing superior multiobjective sequential transfer optimization algorithms. The source code is available at https://github.com/XmingHsueh/MSTOP.

Integrating personalization into federated learning is crucial for addressing data heterogeneity and surpassing the limitations of a single aggregated model. Personalized federated learning excels at capturing inter-client similarities and meeting diverse client needs through custom-made models. However, even with personalized approaches, it’s essential to aggregate knowledge among clients to ensure universal benefits. This paper proposes Federated Dual Objectives and Dual Models (FedDodm), a novel approach that employs two independent models to separately address explicit personalization and implicit generalization objectives in personalized federated learning. By treating these objectives as distinct loss functions and training models accordingly, we achieve a balance between the two through a fusion method. Extensive experiments across various models and learning tasks demonstrate that FedDodm outperforms state-of-the-art federated learning approaches, marking a significant advancement in effectively integrating personalized and generalized knowledge.

The competitive swarm optimizer (CSO) classifies swarm particles into loser and winner particles and then uses the winner particles to efficiently guide the search of the loser particles. This approach has very promising performance in solving large-scale multiobjective optimization problems (LMOPs). However, most studies of CSOs ignore the evolution of the winner particles, although their quality is very important for the final optimization performance. Aiming to fill this research gap, this article proposes a new neural net-enhanced CSO for solving LMOPs, called NN-CSO, which not only guides the loser particles via the original CSO strategy, but also applies our trained neural network (NN) model to evolve winner particles. First, the swarm particles are classified into winner and loser particles by the pairwise competition. Then, the loser particles and winner particles are, respectively, treated as the input and desired output to train the NN model, which tries to learn promising evolutionary dynamics by driving the loser particles toward the winners. Finally, when model training is complete, the winner particles are evolved by the well-trained NN model, while the loser particles are still guided by the winner particles to maintain the search pattern of CSOs. To evaluate the performance of our designed NN-CSO, several LMOPs with up to ten objectives and 1000 decision variables are adopted, and the experimental results show that our designed NN model can significantly improve the performance of CSOs and shows some advantages over several state-of-the-art large-scale multiobjective evolutionary algorithms as well as over model-based evolutionary algorithms.

Evolutionary neural architecture search (ENAS) methods have been successfully used to design convolutional neural network (CNN) architectures automatically. These methods have achieved excellent performance in creating a specific neural architecture for a single task but are less efficient for multiple tasks. Existing ENAS frameworks always repeatedly perform the search from scratch for each task, even though these tasks may be solved by similar CNN architectures. This work presents an evolutionary multi-task convolutional neural architecture search (MTNAS) framework to enable efficient architecture searches in multi-task scenarios by incorporating architectural similarities. The proposed MTNAS constructs architectures for different tasks simultaneously by implementing a knowledge-sharing mechanism among multiple search processes. Specifically, promising architectures found in one search process can be transferred and reused to generate high-quality architectures for others. Furthermore, we devise an adaptive strategy to dynamically adjust the frequency of knowledge transfer, aiming to alleviate the potential effect of negative transfer. Extensive experiments demonstrate that MTNAS can outperform state-of-the-art NAS methods or achieve comparable performance in different tasks but with 2× less search cost.

In recent years, surrogate-assisted evolutionary algorithms (SAEAs) have been sufficiently studied for tackling computationally expensive multiobjective optimization problems (EMOPs), as they can quickly estimate the qualities of solutions by using surrogate models to substitute for expensive evaluations. However, most existing SAEAs only show promising performance for solving EMOPs with no more than 10 dimensions, and become less efficient for tackling EMOPs with higher dimensionality. Thus, this article proposes a new SAEA with a simplified helper task for tackling high-dimensional EMOPs. In each generation, one simplified task will be generated artificially by using random dimension reduction on the target task (i.e., the target EMOPs). Then, two surrogate models are trained for the helper task and the target task, respectively. Based on the trained surrogate models, evolutionary multitasking optimization is run to solve these two tasks so that the experiences of solving the helper task can be transferred to speed up the convergence of tackling the target task. Moreover, an effective model management strategy is designed to select new promising samples for training the surrogate models. When compared to five competitive SAEAs on four well-known benchmark suites, the experiments validate the advantages of the proposed algorithm on most test cases.

Evolutionary transfer optimization (ETO) has been becoming a hot research topic in the field of evolutionary computation, which is based on the fact that knowledge learning and transfer across the related optimization exercises can improve the efficiency of others. However, rare studies employ ETO to solve large-scale multiobjective optimization problems (LMOPs). To fill this research gap, this article proposes a new multitasking ETO algorithm via a powerful transfer learning model to simultaneously solve multiple LMOPs. In particular, inspired by adversarial domain adaptation in transfer learning, a discriminative reconstruction network (DRN) model (containing an encoder, a decoder, and a classifier) is created for each LMOP. At each generation, the DRN is trained by the currently obtained nondominated solutions for all LMOPs via backpropagation with gradient descent. With this well-trained DRN model, the proposed algorithm can transfer the solutions of source LMOPs directly to the target LMOP for assisting its optimization, can evaluate the correlation between the source and target LMOPs to control the transfer of solutions, and can learn a dimensional-reduced Pareto-optimal subspace of the target LMOP to improve the efficiency of transfer optimization in the large-scale search space. Moreover, we propose a real-world multitasking LMOP suite to simulate the training of deep neural networks (DNNs) on multiple different classification tasks. Finally, the effectiveness of the proposed algorithm has been validated in this real-world problem suite and the other two synthetic problem suites.

Recent decades have witnessed great advancements in multiobjective evolutionary algorithms (MOEAs) for multiobjective optimization problems (MOPs). However, these progressively improved MOEAs have not necessarily been equipped with scalable and learnable problem-solving strategies for new and grand challenges brought by the scaling-up MOPs with continuously increasing complexity from diverse aspects, mainly, including expensive cost of function evaluations, many objectives, large-scale search space, time-varying environments, and multitask. Under different scenarios, divergent thinking is required in designing new powerful MOEAs for solving them effectively. In this context, research studies on learnable MOEAs with machine learning techniques have received extensive attention in the field of evolutionary computation. This article begins with a general taxonomy of scaling-up MOPs and learnable MOEAs, followed by an analysis of the challenges that these MOPs pose to traditional MOEAs. Then, we synthetically overview recent advances of learnable MOEAs in solving various scaling-up MOPs, focusing primarily on four attractive directions (i.e., learnable evolutionary discriminators for environmental selection, learnable evolutionary generators for reproduction, learnable evolutionary evaluators for function evaluations, and learnable evolutionary transfer modules for sharing or reusing optimization experience). The insight of learnable MOEAs is offered to readers as a reference to the general track of the efforts in this field.

Most existing evolutionary search strategies are not so efficient when directly handling the decision space of large-scale multiobjective optimization problems (LMOPs). To enhance the efficiency of tackling LMOPs, this article proposes an accelerated evolutionary search (AES) strategy. Its main idea is to learn a gradient-descent-like direction vector (GDV) for each solution via the specially trained feedforward neural network, which may be the learnt possibly fastest convergent direction to reproduce new solutions efficiently. To be specific, a multilayer perceptron (MLP) with only one hidden layer is constructed, in which the number of neurons in the input and output layers is equal to the dimension of the decision space. Then, to get appropriate training data for the model, the current population is divided into two subsets based on the nondominated sorting, and each poor solution in one subset with worse convergence will be paired to an elitist solution in another subset with the minimum angle to it, which is considered most likely to guide it with rapid convergence. Next, this MLP is updated via backpropagation with gradient descent by using the above elaborately prepared dataset. Finally, an accelerated large-scale multiobjective evolutionary algorithm (ALMOEA) is designed by using AES as a reproduction operator. Experimental studies validate the effectiveness of the proposed AES when handling the search space of LMOPs with dimensionality ranging from 1000 to 10000. When compared with six state-of-the-art evolutionary algorithms, the experimental results also show the better efficiency and performance of the proposed optimizer in solving various LMOPs.

Evolutionary large-scale multiobjective optimization (ELMO) has received increasing attention in recent years. This study has compared various existing optimizers for ELMO on different benchmarks, revealing that both benchmarks and algorithms for ELMO still need significant improvement. Thus, a new test suite and a new optimizer framework are proposed to further promote the research of ELMO. More realistic features are considered in the new benchmarks, such as mixed formulation of objective functions, mixed linkages in variables, and imbalanced contributions of variables to the objectives, which are challenging to the existing optimizers. To better tackle these benchmarks, a variable group-based learning strategy is embedded into the new optimizer framework for ELMO, which significantly improves the quality of reproduction in large-scale search space. The experimental results validate that the designed benchmarks can comprehensively evaluate the performance of existing optimizers for ELMO and the proposed optimizer shows distinct advantages in tackling these benchmarks.

The performance of traditional multiobj ective evolutionary algorithms (MOEAs) often deteriorates rapidly when using them to solve large-scale multiobjective optimization problems (LMOPs). To effectively handle LMOPs, we propose a large-scale MOEA via self-guided problem transformation. In the proposed optimizer, the original large-scale search space is transferred to a lower-dimensional weighted space by the guidance of solutions themselves, aiming to effectively search in the weighted space for speeding up the convergence of the population. Specifically, the variables of the target LMOP are adaptively and randomly divided into multiple equal groups, and then solutions are self-guided to construct the small-scale weighted space correspondingly to these variable groups. In this way, each solution is projected as a self-guided vector with multiple weight variables, and then new weight vectors can be generated by searching in the weighted space. Next, new offspring is produced by inversely mapping the newly generated weight vectors to the original search space of this LMOP. Finally, the proposed optimizer is tested on two different LMOP test suites by comparing them with five competitive large-scale MOEAs. Experimental results show some advantages of the proposed algorithm in solving the considered benchmarks.

Performance of multi/many-objective evolutionary algorithms (MOEAs) based on decomposition is highly impacted by the Pareto front (PF) shapes of multi/many-objective optimization problems (MOPs), as their adopted weight vectors may not properly fit the PF shapes. To avoid this mismatch, some MOEAs treat solutions as weight vectors to guide the evolutionary search, which can adapt to the target MOP's PF automatically. However, their performance is still affected by the similarity metric used to select weight vectors. To address this issue, this article proposes a fuzzy decomposition-based MOEA. First, a fuzzy prediction is designed to estimate the population's shape, which helps to exactly reflect the similarities of solutions. Then, N least similar solutions are extracted as weight vectors to obtain N constrained fuzzy subproblems ( N is the population size), and accordingly, a shared weight vector is calculated for all subproblems to provide a stable search direction. Finally, the corner solution for each of m least similar subproblems ( m is the objective number) is preserved to maintain diversity, while one solution having the best aggregated value on the shared weight vector is selected for each of the remaining subproblems to speed up convergence. When compared to several competitive MOEAs in solving a variety of test MOPs, the proposed algorithm shows some advantages at fitting their different PF shapes.

Large-scale multiobjective optimization problems (LMOPs) bring significant challenges for traditional evolutionary operators, as their search capability cannot efficiently handle the huge decision space. Some newly designed search methods for LMOPs usually classify all variables into different groups and then optimize the variables in the same group with the same manner, which can speed up the population's convergence. Following this research direction, this article suggests a differential evolution (DE) algorithm that favors searching the variables with higher importance to the solving of LMOPs. The importance of each variable to the target LMOP is quantized and then all variables are categorized into different groups based on their importance. The variable groups with higher importance are allocated with more computational resources using DE. In this way, the proposed method can efficiently generate offspring in a low-dimensional search subspace formed by more important variables, which can significantly speed up the convergence. During the evolutionary process, this search subspace for DE will be expanded gradually, which can strike a good balance between exploration and exploitation in tackling LMOPs. Finally, the experiments validate that our proposed algorithm can perform better than several state-of-the-art evolutionary algorithms for solving various benchmark LMOPs.

Generally, decomposition-based evolutionary algorithms in many-objective optimization (MaOEA/Ds) have widely used reference vectors (RVs) to provide search directions and maintain diversity. However, their performance is highly affected by the matching degree on the shapes of the RVs and the Pareto front (PF). To address this problem, this article proposes a self-guided RV (SRV) strategy for MaOEA/Ds, aiming to extract RVs from the population using a modified k -means clustering method. To give a promising clustering result, an angle-based density measurement strategy is used to initialize the centroids, which are then adjusted to obtain the final clusters, aiming to properly reflect the population's distribution. Afterward, these centroids are extracted to obtain adaptive RVs for self-guiding the search process. To verify the effectiveness of this SRV strategy, it is embedded into three well-known MaOEA/Ds that originally use the fixed RVs. Moreover, a new strategy of embedding SRV into MaOEA/Ds is discussed when the RVs are adjusted at each generation. The simulation results validate the superiority of our SRV strategy, when tackling numerous many-objective optimization problems with regular and irregular PFs.

Competitive swarm optimizers (CSOs) have shown very promising search efficiency in large-scale decision space. However, they face difficulties when solving large-scale multi-/many-objective optimization problems (LMOPs), as their winner particles are selected by random pairwise competition based on only a single evaluation criterion, which does not provide diverse guidance for LMOPs. To alleviate this issue, this article proposes a comprehensive competitive learning (CCL) strategy for CSOs using three competition mechanisms to guide the particle search. Specifically, environmental competition classifies winner and loser particles from the swarm, while cognitive competition and social competition select one winner particle as the cognitive component and the social component, respectively, to guide the search for loser particles. This competitive learning strategy aims to enhance the search capability of loser particles and provides diverse search directions for solving LMOPs. When compared with eight competitive optimizers, the experimental results validate the high efficiency and effectiveness of our method in solving nine LMOPs with 2–10 objectives and 100–5000 variables.

This paper suggests a novel clustering-based evolutionary algorithm for many-objective optimization problems. Its main idea is to classify the population into a number of clusters, which is expected to solve the difficulty of balancing convergence and diversity in high-dimensional objective space. The individuals showing high similarities on the vector angles are gathered into the same cluster, such that the population’s distribution can be well portrayed by the clusters. To efficiently find these clusters, partitional clustering is first used to classify the union population into${m}$main clusters based on the${m}$axis vectors (${m}$is the number of objectives), and then hierarchical clustering is further run on these${m}$main clusters to get${N}$final clusters (${N}$is the population size and${N>m}$). At last, in environmental selection, one individual from each of${N}$clusters closest to the axis vectors is selected to maintain diversity, while one individual from each of the other clusters is preferred by a simple convergence indicator to ensure convergence. When tackling some well-known test problems with 5–15 objectives, extensive experiments validate the superiority of our algorithm over six competitive many-objective EAs, especially on problems with incomplete and irregular Pareto-optimal fronts.

Recently, it was found that most multiobjective particle swarm optimizers (MOPSOs) perform poorly when tackling many-objective optimization problems (MaOPs). This is mainly because the loss of selection pressure that occurs when updating the swarm. The number of nondominated individuals is substantially increased and the diversity maintenance mechanisms in MOPSOs always guide the particles to explore sparse regions of the search space. This behavior results in the final solutions being distributed loosely in objective space, but far away from the true Pareto-optimal front. To avoid the above scenario, this paper presents a balanceable fitness estimation method and a novel velocity update equation, to compose a novel MOPSO (NMPSO), which is shown to be more effective to tackle MaOPs. Moreover, an evolutionary search is further run on the external archive in order to provide another search pattern for evolution. The DTLZ and WFG test suites with 4-10 objectives are used to assess the performance of NMPSO. Our experiments indicate that NMPSO has superior performance over four current MOPSOs, and over four competitive multiobjective evolutionary algorithms (SPEA2-SDE, NSGA-III, MOEA/DD, and SRA), when solving most of the test problems adopted.