Consensus adjustment is essential in group decision-making (GDM) to reduce the judgment bias of decision-makers (DMs). Advances in communication technology enable greater online participation, foster complex social networks, and provide more diverse evaluation information. However, few existing studies on large-scale social network-based GDM explored consensus adjustment from the perspective of DM interactions. To address this gap, this article proposes a novel large-scale social network GDM method that incorporates consensus adjustment considering the interactions among DMs and subgroups. First, a two-stage semisupervised fuzzy C-means clustering method based on information fusion is introduced to handle the complexity of social networks. Then, to deal with the issue of inconsistent minimum adjustment amounts between individuals, subgroups, and the overall group, a two-stage cross-allocation approach is proposed, which considers the favorable allocation preferences of all DMs under the overall minimum adjustments. Finally, the effectiveness of the proposed method is verified through a case study that demonstrates its practical applicability and performance in achieving consensus in complex social network environments.

Inspired by neuronal diversity in the biological neural system, a plethora of studies proposed to design novel types of artificial neurons and introduce neuronal diversity into artificial neural networks. Recently proposed quadratic neuron, which replaces the inner-product operation in conventional neurons with a quadratic one, have achieved great success in many essential tasks. Despite the promising results of quadratic neurons, there is still an unresolved issue: Is the superior performance of quadratic networks simply due to the increased parameters or due to the intrinsic expressive capability? Without clarifying this issue, the performance of quadratic networks is always suspicious. Additionally, resolving this issue is reduced to finding killer applications of quadratic networks. In this paper, with theoretical and empirical studies, we show that quadratic networks enjoy parametric efficiency, thereby confirming that the superior performance of quadratic networks is due to the intrinsic expressive capability. This intrinsic expressive ability comes from that quadratic neurons can easily represent nonlinear interaction, while it is hard for conventional neurons. Theoretically, we derive the approximation efficiency of quadratic networks over conventional ones in terms of real space and manifolds. Moreover, from the perspective of the Barron space, we demonstrate that there exists a functional space whose functions can be approximated by quadratic networks in a dimension-free error, but the approximation error of conventional networks is dependent on dimensions. Empirically, experimental results on synthetic data, classic benchmarks, and real-world applications show that quadratic models broadly enjoy parametric efficiency, and the gain of efficiency depends on the task.

Distribution linguistic preference relations (DLPRs) play a crucial role in group decision making due to their ability to capture hesitation and uncertainty in individual judgments. By utilizing multiple linguistic variables with associated distribution proportions, DLPRs offer a flexible way to represent preferences. However, current models that use DLPRs often overlook two crucial factors: the ordinal consistency of preference relations and the fairness of adjustment allocation within the DLPRs-based consensus reaching process. In this article, we propose a cooperative game-based minimum adjustment consensus reaching mechanism that accounts for both ordinal consistency and the hesitant degree in DLPRs. This approach leverages the properties of indices in cooperative game theory to ensures a fair allocation of consistency and consensus adjustments, while maintaining ordinal consistency and controlling the hesitant degree of DLPRs through the construction of appropriate constraints to preserve their quality. In addition, a new algorithm is developed to manage completeness, ordinal and acceptable cardinal consistency, consensus-reaching, and hesitation in scenarios involving incomplete DLPRs. Finally, a case study is provided to demonstrate the practical application of the proposed method. Sensitivity and comparative analyzes with existing models are performed to assess the performance of the approach in terms of quality, fairness, and efficiency.

The ability of the minimum cost consensus model (MCCM) to promote consensus reaching in the domain of group decision-making (GDM) has been extensively studied. Recently, the MCCM has been enhanced by introducing the consensus principle and tolerance level to achieve a soft consensus. However, the potential impact of asymmetric and uncertain unit adjustment costs on the effectiveness of the consensus reaching process (CRP) has been overlooked. This paper aims to investigate the implications of uncertain asymmetric costs for achieving consensus with a certain level of tolerance, where new robust MCCMs with uncertain asymmetric costs are constructed under four uncertainty sets for the unit adjustment costs. Considering the linear uncertain-constrained tolerance level and consensus level, we incorporate the insight of an expert with a cost-free threshold into models. Additionally, through a pollutant emission application, the proposed robust MCCMs are able to effectively handle uncertainties arising from costs and improve the quality of the CRP compared to the traditional models. Finally, we conduct simulation experiments and sensitivity analysis to illustrate the effectiveness of the proposed models on achieving a consensus by identifying appropriate parameters.

Abstract. This paper investigates the convergence properties and applications of the three-operator splitting method, also known as the Davis–Yin splitting (DYS) method, integrated with extrapolation and plug-and-play (PnP) denoiser within a nonconvex framework. We first propose an extrapolated DYS method to effectively solve a class of structural nonconvex optimization problems that involve minimizing the sum of three possibly nonconvex functions. Our approach provides an algorithmic framework that encompasses both extrapolated forward–backward splitting and extrapolated Douglas–Rachford splitting methods. To establish the convergence of the proposed method, we rigorously analyze its behavior based on the Kurdyka–Łojasiewicz property, subject to some tight parameter conditions. Moreover, we introduce two extrapolated PnP-DYS methods with convergence guarantee, where the traditional regularization step is replaced by a gradient step–based denoiser. This denoiser is designed using a differentiable neural network and can be reformulated as the proximal operator of a specific nonconvex functional. We conduct extensive experiments on image deblurring and image superresolution problems, where our numerical results showcase the advantage of the extrapolation strategy and the superior performance of the learning-based model that incorporates the PnP denoiser in terms of achieving high-quality recovery images.