Existing multi-objective particle swarm optimisation (MOPSO) algorithms often face bottlenecks when handling complex frontiers, such as gene loss caused by truncation of a single archive and search imbalance resulting from parameter stagnation. To address these challenges, this paper proposes a novel algorithm (CDAMOPSO) based on capacity-driven dual archiving and multi-source feedback coordination. The algorithm innovatively constructs an “overflow-mutation-refill” dual-repository mechanism, which transfers congested solutions overflowing from the main repository to the auxiliary repository for differential activation. This effectively reawakens dormant genes and breaks through population evolutionary stagnation. Concurrently, topological features such as population congestion and dispersion are extracted as feedback signals, and an adaptive closed-loop control law for parameters is designed to achieve dynamic, on-demand allocation of computational resources between global exploration and local exploitation. Furthermore, by introducing a sparse-grid-guided “vanguard-main force” hierarchical strategy, the algorithm utilises asymmetric collaboration among heterogeneous particles to accurately map complex Pareto fronts, including discontinuous and long-tail fronts. Comparative experiments based on 22 standard test sets and 10 mainstream algorithms confirm that CDAMOPSO achieves significant statistical advantages in both convergence accuracy (IGD) and comprehensive coverage (HV), demonstrating outstanding performance in solving complex multi-objective problems.

Addressing the limitations of traditional multi-objective particle swarm optimization (MOPSO) in achieving coordinated optimization of convergence efficiency and solution set distribution, this paper proposes an improved algorithm (MCMOPSO). This algorithm integrates a dual-dimensional density-morphology profile maintenance method with a parameter adjustment mechanism based on a dynamic tracking matrix. Its innovations are concentrated in three aspects: constructing an external density-morphology profile maintenance scheme to achieve precise screening of non-dominated solutions; designing a dynamic tracking matrix recognition system to finely regulate particle trajectories; and establishing a percentile dominance framework to efficiently guide the population toward the Pareto optimal frontier (PF). Comparative experiments were conducted between this algorithm and 10 classical algorithms on three benchmark test problems (ZDT, UF, DTLZ). Performance was quantitatively analyzed using metrics such as hyper-volume (HV) and inverse generational distance (IGD). Results demonstrate that MCMOPSO exhibits superior convergence, distribution uniformity, and stability across most test problems, providing an efficient and reliable solution for complex multi-objective optimization.

In this paper, we introduce two new structured tensors, which are called generalized nonnegative tensors and GM $G\mathcal{M}$ -tensors. These concepts extend the concepts of nonnegative tensors and M $\mathcal{M}$ -tensors. An improved Ky Fan theorem for tensors is presented based on the properties of generalized nonnegative tensors.

In Multi-objective particle swarm optimization (MOPSO), the external archive largely determines how well convergence and diversity are balanced. Ineffective archive maintenance may lead to uneven solution distributions, inaccurate convergence, and premature trapping in local regions.To overcome these limitations, this paper proposes TAMOPSO, a Two-stage Archive Maintenance-based Multi-Objective Particle Swarm Optimization algorithm. In the first stage, adaptive grids with dynamic boundary expansion are used to locate high-density regions. In the second stage, solutions in these regions are evaluated by integrating angle-based diversity assessment and a dual-distance convergence metric, with selection preferences adaptively adjusted according to the evolutionary stage, thereby improving the distribution and Pareto-front coverage of the obtained solution set while controlling archive size. To further enhance particle guidance, a bounded auxiliary archive is introduced to reuse historical high-quality non-dominated solutions discarded during archive maintenance and assist personal best updates. In addition, a stagnation detection-based particle reconstruction strategy is designed, using sparsely distributed elite solutions from the external archive as reconstruction templates to guide stagnant particles back to promising search regions and enhance global exploration. Tests on representative benchmark suites indicate that TAMOPSO produces higher-quality approximation sets than the compared mainstream algorithms in most cases.

The multi-objective particle swarm optimization (MOPSO) is an optimization technique that mimics the foraging behavior of birds to solve difficult optimization problems. MOPSO is well known for its strong global search capability, which efficiently locates solutions that are close to the global optimum across a wide search domain. However, similar to many other optimization algorithms, the fast convergence property of MOPSO can occasionally lead to the population entering the local optimum too soon, obstructing researchers from investigating more efficient solutions. To address this challenge, the study proposes a novel framework that integrates the fireworks algorithm (FA) into MOPSO and establishes a size-double archiving mechanism to maintain population diversity. By preventing population homogenization, this mechanism promotes the retention of better solutions. Additionally, by fusing evolutionary data analysis with particle information, the study offers new individual optimal choices and adaptive parameter tuning to improve the algorithm’s robustness and adaptability and better manage the complexity of multi-objective optimization problems (MOPs). The suggested algorithm is compared with several existing MOPSOs and multi-objective evolutionary algorithms (MOEAs) in simulation experiments. Standard test problems like ZDT, UF, and DTLZ are used in the experiments. The new algorithm performs exceptionally well in terms of improving convergence and population diversity, as well as demonstrating significant competitiveness for solving MOPs.

As an innovative methodology in data processing and knowledge representation, granular-ball computing (GBC) adaptively generates distinct neighborhoods for individual objects, thereby improving both generality and flexibility. By replacing point inputs with granular-balls (GBs), GBC achieves substantial efficiency gains. However, traditional GB-based classifiers may produce unreliable classifications under uncertain conditions. To address this limitation, we propose a novel approach that integrates three-way decision (3WD) theory with GBC, enabling robust handling of uncertain classification problems. This study first introduces a sequential three-way decision with fuzzy granular-ball rough sets (S3WD-FGBRS). We systematically analyze the changing rules of the multilevel decision cost in S3WD-FGBRS and its three regions. Building upon the principle of justifiable granularity, we develop a cost-sensitive three-way granular-ball generation method (CS3W-GBG) based on S3WD-FGBRS that incorporates a granularity optimization mechanism. To validate our approach, we conduct comprehensive experiments using three state-of-the-art GB classifiers and two benchmark classifiers on 12 publicly available datasets. Experimental results demonstrate that CS3W-GBG exhibits strong resilience in processing uncertain data through its 3WD strategy. Furthermore, our method achieves competitive performance compared to existing approaches in terms of classification accuracy and robustness.

In this paper, we propose Pareto Z-eigenvalue inclusion intervals of tensor Z-eigenvalue complementarity problems and the corresponding sufficient criteria for the copositivity of tensors, which only depend on the negative entries and Z-diagonal entries of tensors. Based on these inclusion intervals, upper and lower bounds for the Pareto Z-eigenvalues of tensors are presented. The relationships between these localization sets are obtained. Finally, some sufficient conditions for the solution existence of the TCP and TGEiP are presented, and these conditions are also used to test the vacuum stability of a general scalar potential.

New Pareto eigenvalue inclusion intervals for tensors are proposed to provide some checkable sufficient criteria for the copositivity of tensors, which are depended only on the negative and diagonal entries of a tensor. The relationships among these localization intervals are also obtained. In addition, some sufficient conditions for the solution existence of the TCP and TGEiP are presented, and these conditions are also used to test the vacuum stability of a general scalar potential. The last application is to obtain an upper bound for the coclique number of an uniform hypergraph. Finally, numerical experiments are reported to show the efficiency of the given localization intervals.

Granular-ball computing (GBC) proposed by Xia adaptively generates a different neighborhood for each object, resulting in greater generality and flexibility. Moreover, GBC greatly improves the efficiency by replacing point input with granular-ball. However, the current GB-based classifiers rigidly assign a specific class label to each data instance and lacks of the necessary strategies to address uncertain instances. These far-fetched certain classification approachs toward uncertain instances may suffer considerable risks. In this article, we introduce three-way decision (3WD) into GBC to construct a novel three-way decision with fuzzy granular-ball rough sets (3WD-FGBRS) from the perspective of uncertainty. This helps to construct reasonable multigranularity spaces for handling complex decision problems with uncertainty. First, 3WD-FGBRS is constructed in a data-driven method based on fuzziness, which avoids the subjective definition of certain risk parameters when calculating the threshold pairs. Based on 3WD-FGBRS, we further propose a sequential three-way decision with fuzzy granular-ball rough sets (S3WD-FGBRS) and analyze the fuzziness loss of multilevel decision result in S3WD-FGBRS. Then, the optimal granular-ball space selection mechanism of S3WD-FGBRS is introduced by combining fuzziness and granular-ball space distance. Finally, extensive comparative experiments are conducted with 3 state-of-the-art GB-based classifiers and 3 classical machine learning classifiers on 12 public benchmark datasets. The results show that our models almost outperform other comparison methods in terms of effectiveness, efficiency and robustness.

In this paper, improved singular value inclusion sets for rectangular tensors are given, which are tighter than those in (Zhao and Li in Linear Multilinear Algebra 66(7):1333–1350, 2018). Numerical experiments are reported to show the efficiency of the given inclusion sets.

In this paper, a multi-objective particle swarm optimization algorithm (DEMOPSO) is proposed, which introduces a double elite selection mechanism to select high-quality elite particles in the archive and enhances the convergence and diversity of the population. In addition, the algorithm adjusts the degree of variation according to the particle crowding, which further enhances the diversity of the population and avoids the dilemma of local optimal solutions. Finally, by comparing and analyzing the results with the selected six classical comparison algorithms on the ZDT series of test functions, the experimental results verify that DEMOPSO performs well in terms of convergence and diversity and is able to achieve better optimization results in complex multi-objective optimization problems, demonstrating its superior performance and advantages.

. The positive definiteness of even-order tensors and asymptotically stability of time-invariant polynomial systems are usually determined by the Zeigenvalue inclusion intervals for tensors. In this paper, exploiting the structure of tensors, we obtain some modified Z-eigenvalue inclusion intervals for tensors. As applications, a checkable sufficient condition for the positive definiteness of even-order tensors is presented. Based on the Lyapunov stability theorem and positive definiteness of tensors, the stability of the time-invariant polynomial system is also discussed.

In this paper, we propose a novel multi-objective particle swarm optimization algorithm with a task allocation and archive-guided mutation strategy (TAMOPSO), which effectively solves the problem of inefficient search in traditional algorithms by assigning different evolutionary tasks to particles with different characteristics. First, TAMOPSO divides multiple subpopulations according to the particle distribution status of each iteration of the population and designs a new task allocation mechanism to improve the evolutionary search efficiency. Second, TAMOPSO adopts an adaptive Lévy flight strategy according to the population growth rate, automatically increasing the global variation probability to expand the search range when the population converges and enhancing the local variation to conduct fine search when the population disperses to realize the dynamics of global and local variations. Finally, TAMOPSO measures the contribution of particles to the population optimization through the particle evolution contribution rate index and filters out valuable historical solutions for subsequent reuse to accelerate the convergence speed; in addition, TAMOPSO improves the individual optimal particle selection mechanism, changes the bias of the traditional algorithm, ensures that each particle has an equal opportunity, and enhances the fairness of the selection process. The fairness of the selection process is enhanced at the same time. The performance of TAMOPSO is compared with ten existing algorithms on 22 standard test problems, and the experimental results show that TAMOPSO outperforms the other algorithms in several standard test problems and has better performance in solving multi-objective problems.

In multi-objective particle swarm optimization (MOPSO), challenges persist, including low diversity in external archives, ambiguous individual optimal choice mechanisms, high sensitivity to parameter settings, and the arduous task of balancing global exploration and local exploitation capabilities. To address these issues, this paper introduces a novel multi-objective particle swarm optimization algorithm named HCRMOPSO. The proposed algorithm innovatively leverages hierarchical clustering based on Ward’s linkage to generate the center of mass as reference points, which are then combined with the ideal point and crowding distance. This effectively maintains the external archive, thereby resolving the diversity deficiency commonly found in traditional MOPSO archives. Additionally, HCRMOPSO fuses multiple particles to update the personal best positions. It also adaptively tunes the flight parameters according to the diversity information within each particle’s neighborhood, enhancing the algorithm’s adaptability. Notably, a new strategy is designed for two specific types of particles, further optimizing the search process. The performance of HCRMOPSO is rigorously evaluated against ten existing algorithms on 22 standard test problems. Experimental results demonstrate that HCRMOPSO outperforms its counterparts on multiple benchmarks, showcasing superior effectiveness in handling multi-objective optimization tasks.

In multi-objective particle swarm optimization, achieving a balance between solution convergence and diversity remains a crucial challenge. To cope with this difficulty, this paper proposes a novel multi-objective particle swarm algorithm, called ASDMOPSO, which aims to improve the optimization efficiency through the angular division of the archive and the dynamic update strategy. The algorithm efficiently classifies non-dominated solutions by dividing the external archive region into equal angles, thus achieving fine management and diversity maintenance of solutions during the optimization process. When the external archive overflows, the algorithm removes the solution in the highest density region using the congestion distance metric. At the same time, the research presents a multi-stage initialization approach. This method splits the random population into two subpopulations. Subsequently, a genetic algorithm and a differential evolutionary algorithm are utilized for optimization purposes in each subpopulation, respectively. As a result, the quality of the initial population is enhanced. To explore the solution space more efficiently, this paper designs a dynamic flight parameter adjustment technique. This technique balances exploration and exploitation by adjusting the optimization algorithm parameters in real time. The proposed algorithm is compared with several representative multi-objective optimization algorithms on 22 benchmark functions, and statistical tests, sensitivity analysis, and complexity analysis are conducted. The experimental results show that the ASDMOPSO algorithm is more competitive than other comparison algorithms, with significantly improved optimization efficiency. For example, on the ZDT4 test function, its average IGD value is 0.032, outperforming the standard PSO algorithm and surpassing all other comparison algorithms, thereby validating the algorithm’s superiority in complex multi-objective optimization problems.

The granular-ball (GB)-based classifier exhibits adaptability in creating coarse-grained information granules as input, thereby enhancing its generality and flexibility. Nevertheless, current GB-based classifiers rigidly assign a specific class label to each data instance and lack the necessary strategies to address uncertain instances. Such certain classification approaches to uncertain instances may suffer considerable risks. To solve this problem, We introduced the three-way decision into granular-ball SVM (GBSVM) to construct a robust three-way granularball SVM (3WGBSVM) model for uncertain data, which categorizes data instances into certain classes and uncertain cases. Extensive comparative experiments are conducted with 4 GB-based classifiers on 6 public benchmark datasets. The results show that our model demonstrates robustness in managing uncertain data and effectively mitigates classification risks. Furthermore, our model almost outperforms the other comparative methods in both effectiveness and efficiency.

As a powerful optimization technique, multi-objective particle swarm optimization (MOPSO) has been paid more and more attention by scientists. However, in more complex problems, MOPSO faces the challenges of weak global search ability and easy-to-fall-into local optimality. To address these challenges and obtain better solutions, people have proposed many variants. In this study, a density-guided and adaptive update strategy for multi-objective particle swarm optimization (DAMOPSO) is proposed. First, an adaptive grid is used to determine the mutation particles and guides. Then, the Cauchy mutation operator is performed for the poorly distributed particles to expand the search space of the population. Additionally, the strategy of non-dominated sorting and hyper-region density are devised for maintaining external archives, which contribute to the uniform distribution of optimal solutions. Finally, an adaptive detection strategy based on the adjustment coefficient and conversion efficiency is designed to update the flight parameters. These approaches not only speed up the convergence of algorithms, but also balance exploitation and exploration more effectively. The proposed algorithm is compared with several representative multi-objective optimization algorithms on 22 benchmark functions; meanwhile, statistical tests, ablation experiments, analysis of stability, and complexity are also performed. The experimental results demonstrate DAMOPSO is more competitive than other comparison algorithms. Graphical Abstract

Three-way decision (3WD) theory provides an innovative approach to categorizing uncertain issues into acceptance, rejection, and non-commitment regions. The rough fuzzy sets (RFS) model extends rough sets to handle imprecise or fuzzy concepts. A critical challenge in developing the three-way decision model of rough fuzzy sets (3WDRFS) lies in determining and interpreting threshold pairs. However, the traditional expert-based subjective risk parameters often lead to significant misclassification errors in 3WD. To address this limitation, we propose a novel fuzzy similarity-driven 3WDRFS framework (3WDRFS-FS). Our approach involves two key steps: (1) developing a three-way approximation shadowed set (3WA-SS) model using average-step fuzzy sets (AFS), and (2) formulating an objective function to quantify the fuzzy similarity between AFS and 3WA-SS, which is utilized to derive the optimal threshold pair for the 3WDRFS-FS model. Relevant experimental results validate the effectiveness of our proposed model. Moreover, the 3WDRFS-FS model demonstrates superior performance compared to the 0.5-approximation model (0.5-AM).