In this article, the variational generalized Nash equilibrium (vGNE) seeking problem for general monotonic game with multiple coupling constraints involving dynamical players is explored. Specifically, a distributed vGNE-seeking neural network (vGSNN) with a feedback controller is designed based on high-pass filter, which efficiently transforms players' high-order dynamics into equivalent second-order ones. To further relax the requirement on parameter predesign, we propose a controller that uses adaptive weights to replace the traditional fixed gains, which realizes the full distribution of the vGSNN. Furthermore, to enhance the robustness of the vGSNN against disturbances, a novel sliding-mode controller is incorporated to ensure finite-time disturbance rejection while maintaining the full distribution of the vGSNN. Finally, an uncrewed aerial vehicle (UAV) swarm game is put forward to verify the effectiveness of the vGSNNs.

With the increasing integration of distributed energy resources in modern power grids, efficient and scalable resource allocation strategies are crucial for grid stability and economic operation. To solve such resource allocation, this paper develops a neurodynamic approach based on a heterogeneous linear multi-agent system (MAS) operating under limited sensing and communication capabilities, a scenario frequently encountered in power distribution networks. To address variable-coupled inequality constraints in resource allocation, a smoothing penalty function is introduced to simplify constraint handling. Leveraging fixed-time control theory, the proposed distributed neurodynamic approach with a novel communication mechanism ensures the system state converges to an approximate optimal solution within a fixed time. The effectiveness of the proposed neurodynamic approach is demonstrated through numerical simulations and an economic dispatch problem, highlighting its potential for real-time power grid optimization.

This article proposes new theoretical results on the multiple Mittag-Leffler stability of almost periodic solutions (APOs) for fractional-order delayed neural networks (FDNNs) with nonlinear and nonmonotonic activation functions. Profited from the superior geometrical construction of activation function, the considered FDNNs have multiple APOs with local Mittag-Leffler stability under given algebraic inequality conditions. To solve the algebraic inequality conditions, especially in high-dimensional cases, a distributed optimization (DOP) model and a corresponding neurodynamic solving approach are employed. The conclusions in this article generalize the multiple stability of integer- or fractional-order NNs. Besides, the consideration of the DOP approach can ameliorate the excessive consumption of computational resources when utilizing the LMI toolbox to deal with high-dimensional complex NNs. Finally, a simulation example is presented to confirm the accuracy of the theoretical conclusions obtained, and an experimental example of associative memories is shown.

This article proposes a novel adaptive neurodynamic algorithm (ANA) to seek generalized Nash equilibrium (GNE) of the noncooperative constrained game with different monotone conditions. In the ANA, the adaptive penalty term, which acts as trajectory-dependent penalty parameters, evolves based on the degree of constraints violation until the trajectory enters the action set of noncooperative game. It is shown that the trajectory of the ANA enters the action set in finite time benefited from the adaptive penalty term. Moreover, it is proven that the trajectory exponentially (or polynomially) converges to the unique GNE when the pseudo-gradient of cost function in noncooperative game satisfies strong (or "generalized" strong) monotonicity. To the best of our knowledge, this is the first time to study the polynomial convergence of GNE seeking algorithm. Furthermore, when the pseudo-gradient mentioned above satisfies monotonicity in general, based on Tikhonov regularization method, a new ANA for finding its $\varepsilon $ -generalized Nash equilibrium ( $\varepsilon $ -GNE) is proposed, and the related exponential convergence of the algorithm is established. Finally, the river basin pollution game and 5G base station location game are given as examples to showcase the algorithm's effectiveness.

In this article, a neurodynamic approach based on event-triggered mechanism for solving Lyapunov matrix equation is established. First, employing matrix decomposition technique, the Lyapunov matrix equation is reformulated as a distributed optimization problem. Then, a distributed neurodynamic approach is constructed to solve the corresponding distributed optimization problem owing to its better-parallel computing ability. In order to protect the privacy of agents and fulfill the distributed communication, a primal-dual method with auxiliary variables is introduced. Agents collaborate to solve distributed optimization problem by interacting with auxiliary variables rather than decision variables. Besides, to reduce the communication cost and frequency between agents, the neurodynamic approach incorporates an event-triggered mechanism for Lyapunov matrix equation for the first time. Through theoretical analysis, it is proved that the state solution of the proposed neurodynamic approach converges exponentially and no Zeno behavior occurs. Finally, a numerical example is given to show the feasibility and effectiveness of the proposed event-triggered neurodynamic approach.

During industrial smart manufacturing, many problems can be described as a time-dependent nonlinear equation system (TNES) that needs to be solved cooperatively due to large-scale information flows and transmission lines. In this article, a distributed neurodynamic approach is designed for solving a class of TNESs, over multiagent networks from a distributed optimization perspective. To be specific, the TNES is transformed into a distributed time-varying optimization problem (DTOP) to solve. For reformulated DTOP, the proposed neurodynamic approach has ability to jointly drive all agents to reach consensus while optimizing the global objective function. Furthermore, relying on an effective activated function, the finite-time consensus and fixed-time convergence are proved, and the upper bounds of settling time are given as well. This significantly improves the efficiency of the approach in completing practical engineering tasks. Besides, a real-time approximation method is introduced to solve the inverse of the Hessian matrices of the local objective functions in real time, causing the proposed neurodynamic approach to be inverse-free and more suitable for computing complex problems. It is verified that such a real-time approximation method can still ensure convergence within fixed time. Finally, some numerical examples and a case study of multirobot moving target tracking are given to demonstrate the effectiveness of the proposed approach.

In industrial intelligent manufacturing processes, restricted by resources and transmission lines, many problems can be described as distributed optimization problems that need to be solved cooperatively on the premise of resource sharing, i.e., distributed resource allocation (DRA). In this article, several neurodynamic approaches are proposed to solve multiple constrained DRAs with planned or self-regulated demand over switched communication topologies. In the problems considered, the planned demand is the well-studied fixed resource demand, and the self-regulated demand refers to the scenario where the demand of users may change unpredictably. To solve DRAs with two different demands mentioned above and the multiple coupled constraints distributedly, a finite-time tracking technique and fixed-time projection technology are utilized such that local Lagrange multipliers reach consensus in finite time. It is proved that the proposed neurodynamic approaches are able to address the coupled equality and nonlinear inequality constraints simultaneously, and the states of agents along each neurodynamic approach asymptotically converge to an optimal solution to the related problem. Finally, a smart grid application is provided to verify the effectiveness of the proposed neurodynamic approaches and demonstrate the advantage of self-regulated demand over planned demand.

In this article, an adaptive neurodynamic approach over multiagent systems is designed to solve nonsmooth distributed resource allocation problems (DRAPs) with affine-coupled equality constraints, coupled inequality constraints, and private set constraints. It is to say, agents focus on tracking the optimal allocation to minimize the team cost under more general constraints. Among the considered constraints, multiple coupled constraints are dealt with by introducing auxiliary variables to make Lagrange multipliers reach consensus. Furthermore, aiming to address private set constraints, an adaptive controller is proposed with the aid of the penalty method, thus avoiding the disclosure of global information. Through using the Lyapunov stability theory, the convergence of this neurodynamic approach is analyzed. In addition, to reduce the communication burden of systems, the proposed neurodynamic approach is improved by introducing an event-triggered mechanism. In this case, the convergence property is also explored, and the Zeno phenomenon is excluded. Finally, a numerical example and a simplified problem on a virtual 5G system are implemented to demonstrate the effectiveness of the proposed neurodynamic approaches.

In this article, a distributed optimization algorithm is proposed for solving a distributed resource allocation problem (DRAP) with general inequality and heterogeneous coupled equality constraints. The communication topologies herein are considered to be jointly connected and directed interacted. To deal with the effects of inequality constraints, an adaptive item of updating penalty gain on-line is introduced in algorithmic design, which enforces the state enter to the constraint sets dynamically. Further, with the aid of Lyapunov method, the convergence to global optimal solution of nonsmooth DRAP is obtained. To effectively alleviate the communication burden caused by frequent interactions, an event-triggered mechanism is proposed to drive the agents with free-initial state, while also ensuring the exclusion of Zeno behavior. Compared with existing algorithms for DRAP, the time-varying auxiliary function designed in distributed algorithms herein avoids the preemptive estimation of global parameters that may cause the failure of the distributed framework, including the global Lipschitz coefficients of objective functions and the eigenvalues of full Laplacian matrix. Finally, numerical simulations and application of economic dispatch in smart grid illustrate the validity of designed algorithm.

In this article, based on the multiagent system with second-order dynamics, two neurodynamic approaches are proposed to solve the distributed optimal allocation problem (DOAP) with equality resource interaction, inequality resource interaction, and local feasible constraints.over the switching communication topologies. To address the equality and inequality resource interactions in a distributed way, the corresponding auxiliary variables are introduced to ensure the local estimations of Lagrangian multipliers reach consensus in a distributed manner. On this basis, a novel generalized second-order neurodynamic approach is presented to solve the nonsmooth DOAP, and the theoretical proof of convergence is provided. Furthermore, to prevent global information from being involved, another generalized second-order neurodynamic approach is designed and its effectiveness is also analyzed. Finally, a numerical example and an application of the maximum network utility problem are simulated to verify the correctness of the conclusions.

In this paper, a noncooperative game with private and coupled constraints under aperiodically intermittent communication is studied, and a distributed generalized Nash equilibrium (GNE) seeking algorithm with two-time-scale structure is proposed to address the issue. Under a practical changing environment that causes aperiodically intermittent communications, players cannot directly and continuously obtain the action information of other players. That is, each player only estimates the actions of all the other players in the communication period, which greatly saves the communication cost on the basis of ensuring actual estimation. At the same time, an adaptive technique is introduced to deal with private constraints, where the penalty parameters can be dynamically adjusted according to the degree of constraint violation. On the basis of players' actions entering the action set, the GNE seeking algorithm is fully distributed. Finally, the effectiveness of the algorithm is verified by economic dispatch game.

This article introduces an adaptive memristor-programming neurodynamic approach (AMPNA) to tackle optimization problems that are nonconvex and nonsmooth with inequality and equality constraints. In the circumstance that requiring neither estimating penalty parameters, nor the coerciveness of inequality constraints, the state of the AMPNA can go into the feasible region from any initial points within a finite amount of time and ultimately converge to the critical point set of the aforementioned optimization problem. Differ from the existing neurodynamic approach (NA), AMPNA has superiority in using memristor. On the one hand, with regard to power consumption, AMPNA makes the most of memristor’s unconventional characteristics to execute within the flux-charge realm. Compared with conventional NA executing within the voltage-current realm, AMPNA executes within the flux-charge realm and consumes power only in the analog transient. Once the analog transient is complete, all voltages, currents and powers in the AMPNA disappear. On the other hand, in terms of result storage, since the memristor has the ability to calculate and save information at the same physical location, the AMPNA no longer needs additional memories, and can implement the calculation scheme by the principle of in-memory computation. Therefore, the AMPNA presented in this article has significant advantages in reducing power consumption and storage space. Finally, AMPNA’s optimization capacity and exceptional performance are confirmed through numerical simulations.

This article provides a novel projection neural network (PNN) for a category of monotone variational inequality (MVI). For simplifying calculation, the feasible region of MVI is separated into two parts: one is the set on which projection has quantitative expression, and the rest part can be described by inequalities. A novel continuous-time PNN is well designed in this article based on the division and its states are verified to exist and ultimately converge to a solution of MVI. The proposed PNN differs from conventional approaches since it only depends on the projection operator on the sets with projection-quantifiable structure instead of the whole feasible region. The feature determines the PNN here owns the advantage of low amount of calculation and flexibility when encountering diverse MVIs. Compared with existing numerical algorithms, model convergence in this article is established on weaker assumptions and one can monitor its state at every moment. Finally, the effectiveness and the efficiency of PNN are testified by several simulation examples and an application in image fusion.

Recent advances and emerging approaches in neural model and learning (i.e. deep neural network) have led to an unparalleled surge of interest in the topic of neural networks. Neural networks provid...

A distributed optimization problem (DOP) with affine equality and convex inequality constraints is studied in this article. First, the consensus constraint of the considered DOP is relaxed and a related approximate DOP (ADOP) is presented. It is proved that the optimal solutions of the ADOP (i.e., the near-optimal solutions of the original DOP) are able to approach the optimal solutions of the original DOP. A continuous-time algorithm is proposed for the ADOP and it is demonstrated that the state solution of the presented algorithm converges to the critical point set of the ADOP with general locally Lipschitz continuous objective functions. This means the presented algorithm is efficient for distributed nonconvex optimization problems. Particularly, when the objective functions are convex ones, the state solution of the presented algorithm is further proved to converge to a near-optimal solution of the original DOP. One illustrative example and an application on load sharing problems are shown to validate the effectiveness of the proposed algorithm.

This paper presents the multistability analysis of almost periodic state solutions for memristive Cohen-Grossberg neural networks (MCGNNs) with both distributed delay and discrete delay. The activation function of the considered MCGNNs is generalized to be nonmonotonic and nonpiecewise linear. It is shown that the MCGNNs with n-neuron have (K + 1)nlocally exponentially stable almost periodic solutions, where nature number K depends on the geometrical structure of the considered activation function. Compared with the previous related works, the number of almost periodic state solutions of the MCGNNs is extensively increased. The obtained conclusions in this paper are also capable of studying the multistability of equilibrium points or periodic solutions of the MCGNNs. Moreover, the enlarged attraction basins of attractors are estimated based on original partition. Some comparisons and convincing numerical examples are provided to substantiate the superiority and efficiency of obtained results.

Complex-variable pseudoconvex optimization has been widely used in numerous scientific and engineering optimization problems. A neurodynamic approach is proposed in this paper for complex-variable pseudoconvex optimization problems subject to bound and linear equality constraints. An efficient penalty function is introduced to guarantee the boundedness of the state of the presented neural network, and make the state enter the feasible region of the considered optimization in finite time and stay there thereafter. The state is also shown to be convergent to an optimal point of the considered optimization. Compared with other neurodynamic approaches, the presented neural network does not need any penalty parameters, and has lower model complexity. Furthermore, some additional assumptions in other existing related neural networks are also removed in this paper, such as the assumption that the objective function is lower bounded over the equality constraint set and so on. Finally, some numerical examples and an application in beamforming formulation are provided.

In this paper, based on calculus and penalty method, a one-layer recurrent neural network is proposed for solving constrained complex-variable convex optimization. It is proved that for any initial point from a given domain, the state of the proposed neural network reaches the feasible region in finite time and converges to an optimal solution of the constrained complex-variable convex optimization finally. In contrast to existing neural networks for complex-variable convex optimization, the proposed neural network has a lower model complexity and better convergence. Some numerical examples and application are presented to substantiate the effectiveness of the proposed neural network.

Memristor, as the future of artificial intelligence, has been widely used in pattern recognition or signal processing from sensor arrays. Memristor-based recurrent neural network (MRNN) is an ideal model to mimic the functionalities of the human brain due to the physical properties of memristor. In this paper, the periodicity for memristor-based Cohen–Grossberg neural networks (MCGNNs) is studied. The neural network (NN) considered in this paper is based on the memristor and involves time-varying delays, distributed delays and impulsive effects. The boundedness and monotonicity of the activation function are not assumed. By some inequality technique and contraction mapping principle, we prove the existence, uniqueness and exponential stability of periodic solution for MCGNNs. Finally, some numeral examples and comparisons are provided to illustrate the validation of our results.

Pseudoconvex optimization problem, as an important nonconvex optimization problem, plays an important role in scientific and engineering applications. In this paper, a recurrent one-layer neural network is proposed for solving the pseudoconvex optimization problem with equality and inequality constraints. It is proved that from any initial state, the state of the proposed neural network reaches the feasible region in finite time and stays there thereafter. It is also proved that the state of the proposed neural network is convergent to an optimal solution of the related problem. Compared with the related existing recurrent neural networks for the pseudoconvex optimization problems, the proposed neural network in this paper does not need the penalty parameters and has a better convergence. Meanwhile, the proposed neural network is used to solve three nonsmooth optimization problems, and we make some detailed comparisons with the known related conclusions. In the end, some numerical examples are provided to illustrate the effectiveness of the performance of the proposed neural network.

This paper presents a neurodynamic optimization approach to bilevel quadratic programming (BQP). Based on the Karush-Kuhn-Tucker (KKT) theorem, the BQP problem is reduced to a one-level mathematical program subject to complementarity constraints (MPCC). It is proved that the global solution of the MPCC is the minimal one of the optimal solutions to multiple convex optimization subproblems. A recurrent neural network is developed for solving these convex optimization subproblems. From any initial state, the state of the proposed neural network is convergent to an equilibrium point of the neural network, which is just the optimal solution of the convex optimization subproblem. Compared with existing recurrent neural networks for BQP, the proposed neural network is guaranteed for delivering the exact optimal solutions to any convex BQP problems. Moreover, it is proved that the proposed neural network for bilevel linear programming is convergent to an equilibrium point in finite time. Finally, three numerical examples are elaborated to substantiate the efficacy of the proposed approach.

This paper presents new results on neurodynamic optimization approach to solve bilevel linear programming problems (BLPPs) with linear inequality constraints. A sub-gradient recurrent neural network is proposed for solving the BLPPs. It is proved that the state convergence time period is finite and can be quantitatively estimated. Compared with existing recurrent neural networks for BLPPs, the proposed neural network does not have any design parameter and can solve the BLPPs in finite time. Some numerical examples are introduced to show the effectiveness of the proposed neural network.

In this paper, a two-layer recurrent neural network is proposed to solve the nonsmooth convex optimization problem subject to convex inequality and linear equality constraints. Compared with existing neural network models, the proposed neural network has a low model complexity and avoids penalty parameters. It is proved that from any initial point, the state of the proposed neural network reaches the equality feasible region in finite time and stays there thereafter. Moreover, the state is unique if the initial point lies in the equality feasible region. The equilibrium point set of the proposed neural network is proved to be equivalent to the Karush-Kuhn-Tucker optimality set of the original optimization problem. It is further proved that the equilibrium point of the proposed neural network is stable in the sense of Lyapunov. Moreover, from any initial point, the state is proved to be convergent to an equilibrium point of the proposed neural network. Finally, as applications, the proposed neural network is used to solve nonlinear convex programming with linear constraints and L1 -norm minimization problems.