Parallel Solution of Nonlinear Projection Equations in a Multitask Learning Framework

Nonlinear projection equations (NPEs) provide a unified framework for addressing various constrained nonlinear optimization and engineering problems. However, when it comes to solving multiple NPEs, traditional numerical integration methods are not efficient enough. This is because traditional methods solve each NPE iteratively and independently. In this article, we propose a novel approach based on multitask learning (MTL) for solving multiple NPEs. The solution procedure is outlined as follows. First, we model each NPE as a system of ordinary differential equations (ODEs) using neurodynamic optimization. Second, for each ODE system, we use a physics-informed neural network (PINN) as the solution. Third, we use a multibranch MTL framework, where each branch corresponds to a PINN model. This allows us to solve multiple NPEs in parallel by training a single neural network model. Experimental results show that our approach has superior computational performance, especially when the number of NPEs to be solved is large.

Supervised Feature Selection via Collaborative Neurodynamic Optimization

As a crucial part of machine learning and pattern recognition, feature selection aims at selecting a subset of the most informative features from the set of all available features. In this article, supervised feature selection is at first formulated as a mixed-integer optimization problem with an objective function of weighted feature redundancy and relevancy subject to a cardinality constraint on the number of selected features. It is equivalently reformulated as a bound-constrained mixed-integer optimization problem by augmenting the objective function with a penalty function for realizing the cardinality constraint. With additional bilinear and linear equality constraints for realizing the integrality constraints, it is further reformulated as a bound-constrained biconvex optimization problem with two more penalty terms. Two collaborative neurodynamic optimization (CNO) approaches are proposed for solving the formulated and reformulated feature selection problems. One of the proposed CNO approaches uses a population of discrete-time recurrent neural networks (RNNs), and the other use a pair of continuous-time projection networks operating concurrently on two timescales. Experimental results on 13 benchmark datasets are elaborated to substantiate the superiority of the CNO approaches to several mainstream methods in terms of average classification accuracy with three commonly used classifiers.

Enhancing neurodynamic approach with physics-informed neural networks for solving non-smooth convex optimization problems

This paper proposes a deep learning approach for solving non-smooth convex optimization problems (NCOPs), which have broad applications in computer science, engineering, and physics. Our approach combines neurodynamic optimization with physics-informed neural networks (PINNs) to provide an efficient and accurate solution. We first use neurodynamic optimization to formulate an initial value problem (IVP) that involves a system of ordinary differential equations for the NCOP. We then introduce a modified PINN as an approximate state solution to the IVP. Finally, we develop a dedicated algorithm to train the model to solve the IVP and minimize the NCOP objective simultaneously. Unlike existing numerical integration methods, a key advantage of our approach is that it does not require the computation of a series of intermediate states to produce a prediction of the NCOP. Our experimental results show that this computational feature results in fewer iterations being required to produce more accurate prediction solutions. Furthermore, our approach is effective in finding feasible solutions that satisfy the NCOP constraint.

A Projection Neural Network to Nonsmooth Constrained Pseudoconvex Optimization

In this article, a single-layer projection neural network based on penalty function and differential inclusion is proposed to solve nonsmooth pseudoconvex optimization problems with linear equality and convex inequality constraints, and the bound constraints, such as box and sphere types, in inequality constraints are processed by projection operator. By introducing the Tikhonov-like regularization method, the proposed neural network no longer needs to calculate the exact penalty parameters. Under mild assumptions, by nonsmooth analysis, it is proved that the state solution of the proposed neural network is always bounded and globally exists, and enters the constrained feasible region in a finite time, and never escapes from this region again. Finally, the state solution converges to an optimal solution for the considered optimization problem. Compared with some other existing neural networks based on subgradients, this algorithm eliminates the dependence on the selection of the initial point, which is a neural network model with a simple structure and low calculation load. Three numerical experiments and two application examples are used to illustrate the global convergence and effectiveness of the proposed neural network.

A collaborative neurodynamic optimization algorithm to traveling salesman problem

This paper proposed a collaborative neurodynamic optimization (CNO) method to solve traveling salesman problem (TSP). First, we construct a Hopfield neural network (HNN) with $$n \times n$$ n × n neurons for the n cities. Second, to ensure the convergence of continuous HNN (CHNN), we reformulate TSP to satisfy the convergence condition of CHNN and solve TSP by CHNN. Finally, a population of CHNNs is used to search for local optimal solutions of TSP and the globally optimal solution is obtained using particle swarm optimization. Experimental results show the effectiveness of the CNO approach for solving TSP.

Sparse signal reconstruction via collaborative neurodynamic optimization

In this paper, we formulate a mixed-integer problem for sparse signal reconstruction and reformulate it as a global optimization problem with a surrogate objective function subject to underdetermined linear equations. We propose a sparse signal reconstruction method based on collaborative neurodynamic optimization with multiple recurrent neural networks for scattered searches and a particle swarm optimization rule for repeated repositioning. We elaborate on experimental results to demonstrate the outperformance of the proposed approach against ten state-of-the-art algorithms for sparse signal reconstruction.

A one-layer recurrent neural network for nonsmooth pseudoconvex optimization with quasiconvex inequality and affine equality constraints

As two important types of generalized convex functions, pseudoconvex and quasiconvex functions appear in many practical optimization problems. The lack of convexity poses some difficulties in solving pseudoconvex optimization with quasiconvex constraint functions. In this paper, we propose a one-layer recurrent neural network for solving such problems. We prove that the state of the proposed neural network is convergent from the feasible region to an optimal solution of the given optimization problem. We show that the proposed neural network has several advantages over the existing neural networks for pseudoconvex optimization. Specifically, the proposed neural network is applicable to optimization problems with quasiconvex inequality constraints as well as affine equality constraints. In addition, parameter matrix inversion is avoided and some assumptions on the objective function and inequality constraints in existing results are relaxed. We demonstrate the superior performance and characteristics of the proposed neural network with simulation results in three numerical examples.

Multivehicle Task Assignment Based on Collaborative Neurodynamic Optimization With Discrete Hopfield Networks

This article presents a collaborative neurodynamic optimization (CNO) approach to multivehicle task assignments (TAs). The original combinatorial quadratic optimization problem for TA is reformulated as a quadratic unconstrained binary optimization (QUBO) problem with a quadratic utility function and a penalty function for handling load capacity and cooperation constraints. In the framework of CNO with a population of discrete Hopfield networks (DHNs), a TA algorithm is proposed for solving the formulated QUBO problem. Superior experimental results in four typical multivehicle operation scenarios are reported to substantiate the efficacy of the proposed neurodynamics-based TA approach.

Cardinality-constrained portfolio selection based on collaborative neurodynamic optimization

Portfolio optimization is one of the most important investment strategies in financial markets. It is practically desirable for investors, especially high-frequency traders, to consider cardinality constraints in portfolio selection, to avoid odd lots and excessive costs such as transaction fees. In this paper, a collaborative neurodynamic optimization approach is presented for cardinality-constrained portfolio selection. The expected return and investment risk in the Markowitz framework are scalarized as a weighted Chebyshev function and the cardinality constraints are equivalently represented using introduced binary variables as an upper bound. Then cardinality-constrained portfolio selection is formulated as a mixed-integer optimization problem and solved by means of collaborative neurodynamic optimization with multiple recurrent neural networks repeatedly repositioned using a particle swarm optimization rule. The distribution of resulting Pareto-optimal solutions is also iteratively perfected by optimizing the weights in the scalarized objective functions based on particle swarm optimization. Experimental results with stock data from four major world markets are discussed to substantiate the superior performance of the collaborative neurodynamic approach to several exact and metaheuristic methods.

Minimax and Biobjective Portfolio Selection Based on Collaborative Neurodynamic Optimization

Portfolio selection is one of the important issues in financial investments. This article is concerned with portfolio selection based on collaborative neurodynamic optimization. The classic Markowitz mean-variance (MV) framework and its variant mean conditional value-at-risk (CVaR) are formulated as minimax and biobjective portfolio selection problems. Neurodynamic approaches are then applied for solving these optimization problems. For each of the problems, multiple neural networks work collaboratively to characterize the efficient frontier by means of particle swarm optimization (PSO)-based weight optimization. Experimental results with stock data from four major markets show the performance and characteristics of the collaborative neurodynamic approaches to the portfolio optimization problems.

Two-timescale neurodynamic approaches to supervised feature selection based on alternative problem formulations

Feature selection is a crucial step in data processing and machine learning. While many greedy and sequential feature selection approaches are available, a holistic neurodynamics approach to supervised feature selection is recently developed via fractional programming by minimizing feature redundancy and maximizing relevance simultaneously. In view that the gradient of the fractional objective function is also fractional, alternative problem formulations are desirable to obviate the fractional complexity. In this paper, the fractional programming problem formulation is equivalently reformulated as bilevel and bilinear programming problems without using any fractional function. Two two-timescale projection neural networks are adapted for solving the reformulated problems. Experimental results on six benchmark datasets are elaborated to demonstrate the global convergence and high classification performance of the proposed neurodynamic approaches in comparison with six mainstream feature selection approaches.

A neurodynamic optimization approach to supervised feature selection via fractional programming

Feature selection is an important issue in machine learning and data mining. Most existing feature selection methods are greedy in nature thus are prone to sub-optimality. Though some global feature selection methods based on unsupervised redundancy minimization can potentiate clustering performance improvements, their efficacy for classification may be limited. In this paper, a neurodynamics-based holistic feature selection approach is proposed via feature redundancy minimization and relevance maximization. An information-theoretic similarity coefficient matrix is defined based on multi-information and entropy to measure feature redundancy with respect to class labels. Supervised feature selection is formulated as a fractional programming problem based on the similarity coefficients. A neurodynamic approach based on two one-layer recurrent neural networks is developed for solving the formulated feature selection problem. Experimental results with eight benchmark datasets are discussed to demonstrate the global convergence of the neural networks and superiority of the proposed neurodynamic approach to several existing feature selection methods in terms of classification accuracy, precision, recall, and F-measure.

A Novel Neurodynamic Approach to Constrained Complex-Variable Pseudoconvex Optimization

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.

A generalized neural network for distributed nonsmooth optimization with inequality constraint

In this paper, a generalized neural network with a novel auxiliary function is proposed to solve a distributed non-differentiable optimization over a multi-agent network. The constructed auxiliary function can ensure that the state solution of proposed neural network is bounded, and enters the inequality constraint set in finite time. Furthermore, the proposed neural network is demonstrated to reach consensus and ultimately converges to the optimal solution under several mild assumptions. Compared with the existing methods, the neural network proposed in this paper has a simple structure with a low amount of state variables, and does not depend on projection operator method for constrained distributed optimization. Finally, two numerical simulations and an application in power system are delineated to show the characteristics and practicability of the presented neural network.

A Collaborative Neurodynamic Approach to Multiobjective Optimization

There are two ultimate goals in multiobjective optimization. The primary goal is to obtain a set of Pareto-optimal solutions while the secondary goal is to obtain evenly distributed solutions to characterize the efficient frontier. In this paper, a collaborative neurodynamic approach to multiobjective optimization is presented to attain both goals of Pareto optimality and solution diversity. The multiple objectives are first scalarized using a weighted Chebyshev function. Multiple projection neural networks are employed to search for Pareto-optimal solutions with the help of a particle swarm optimization (PSO) algorithm in reintialization. To diversify the Pareto-optimal solutions, a holistic approach is proposed by maximizing the hypervolume (HV) using again a PSO algorithm. The experimental results show that the proposed approach outperforms three other state-of-the-art multiobjective algorithms (i.e., HMOEA/D, MOEA/DD, and NSGAIII) most of times on 37 benchmark datasets in terms of HV and inverted generational distance.

A One-Layer Recurrent Neural Network for Pseudoconvex Optimization Problems With Equality and Inequality Constraints

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.

A Collective Neurodynamic Approach to Constrained Global Optimization

Global optimization is a long-lasting research topic in the field of optimization, posting many challenging theoretic and computational issues. This paper presents a novel collective neurodynamic method for solving constrained global optimization problems. At first, a one-layer recurrent neural network (RNN) is presented for searching the Karush-Kuhn-Tucker points of the optimization problem under study. Next, a collective neuroydnamic optimization approach is developed by emulating the paradigm of brainstorming. Multiple RNNs are exploited cooperatively to search for the global optimal solutions in a framework of particle swarm optimization. Each RNN carries out a precise local search and converges to a candidate solution according to its own neurodynamics. The neuronal state of each neural network is repetitively reset by exchanging historical information of each individual network and the entire group. Wavelet mutation is performed to avoid prematurity, add diversity, and promote global convergence. It is proved in the framework of stochastic optimization that the proposed collective neurodynamic approach is capable of computing the global optimal solutions with probability one provided that a sufficiently large number of neural networks are utilized. The essence of the collective neurodynamic optimization approach lies in its potential to solve constrained global optimization problems in real time. The effectiveness and characteristics of the proposed approach are illustrated by using benchmark optimization problems.

A Two-Time-Scale Neurodynamic Approach to Constrained Minimax Optimization

This paper presents a two-time-scale neurodynamic approach to constrained minimax optimization using two coupled neural networks. One of the recurrent neural networks is used for minimizing the objective function and another is used for maximization. It is shown that the coupled neurodynamic systems operating in two different time scales work well for minimax optimization. The effectiveness and characteristics of the proposed approach are illustrated using several examples. Furthermore, the proposed approach is applied for H_∞ model predictive control.

Integration-Enhanced Zhang Neural Network for Real-Time-Varying Matrix Inversion in the Presence of Various Kinds of Noises

Matrix inversion often arises in the fields of science and engineering. Many models for matrix inversion usually assume that the solving process is free of noises or that the denoising has been conducted before the computation. However, time is precious for the real-time-varying matrix inversion in practice, and any preprocessing for noise reduction may consume extra time, possibly violating the requirement of real-time computation. Therefore, a new model for time-varying matrix inversion that is able to handle simultaneously the noises is urgently needed. In this paper, an integration-enhanced Zhang neural network (IEZNN) model is first proposed and investigated for real-time-varying matrix inversion. Then, the conventional ZNN model and the gradient neural network model are presented and employed for comparison. In addition, theoretical analyses show that the proposed IEZNN model has the global exponential convergence property. Moreover, in the presence of various kinds of noises, the proposed IEZNN model is proven to have an improved performance. That is, the proposed IEZNN model converges to the theoretical solution of the time-varying matrix inversion problem no matter how large the matrix-form constant noise is, and the residual errors of the proposed IEZNN model can be arbitrarily small for time-varying noises and random noises. Finally, three illustrative simulation examples, including an application to the inverse kinematic motion planning of a robot manipulator, are provided and analyzed to substantiate the efficacy and superiority of the proposed IEZNN model for real-time-varying matrix inversion.

Echo State Networks With Orthogonal Pigeon-Inspired Optimization for Image Restoration

In this paper, a neurodynamic approach for image restoration is proposed. Image restoration is a process of estimating original images from blurred and/or noisy images. It can be considered as a mapping problem that can be solved by neural networks. Echo state network (ESN) is a recurrent neural network with a simplified training process, which is adopted to estimate the original images in this paper. The parameter selection is important to the performance of the ESN. Thus, the pigeon-inspired optimization (PIO) approach is employed in the training process of the ESN to obtain desired parameters. Moreover, the orthogonal design strategy is utilized in the initialization of PIO to improve the diversity of individuals. The proposed method is tested on several deteriorated images with different sorts and levels of blur and/or noise. Results obtained by the improved ESN are compared with those obtained by several state-of-the-art methods. It is verified experimentally that better image restorations can be obtained for different blurred and/or noisy instances with the proposed neurodynamic method. In addition, the performance of the orthogonal PIO algorithm is compared with that of several existing bioinspired optimization algorithms to confirm its superiority.

A Neurodynamic Approach for Real-Time Scheduling via Maximizing Piecewise Linear Utility

In this paper, we study a set of real-time scheduling problems whose objectives can be expressed as piecewise linear utility functions. This model has very wide applications in scheduling-related problems, such as mixed criticality, response time minimization, and tardiness analysis. Approximation schemes and matrix vectorization techniques are applied to transform scheduling problems into linear constraint optimization with a piecewise linear and concave objective; thus, a neural network-based optimization method can be adopted to solve such scheduling problems efficiently. This neural network model has a parallel structure, and can also be implemented on circuits, on which the converging time can be significantly limited to meet real-time requirements. Examples are provided to illustrate how to solve the optimization problem and to form a schedule. An approximation ratio bound of 0.5 is further provided. Experimental studies on a large number of randomly generated sets suggest that our algorithm is optimal when the set is nonoverloaded, and outperforms existing typical scheduling strategies when there is overload. Moreover, the number of steps for finding an approximate solution remains at the same level when the size of the problem (number of jobs within a set) increases.

Neurodynamics-Based Robust Pole Assignment for High-Order Descriptor Systems

In this paper, a neurodynamic optimization approach is proposed for synthesizing high-order descriptor linear systems with state feedback control via robust pole assignment. With a new robustness measure serving as the objective function, the robust eigenstructure assignment problem is formulated as a pseudoconvex optimization problem. A neurodynamic optimization approach is applied and shown to be capable of maximizing the robust stability margin for high-order singular systems with guaranteed optimality and exact pole assignment. Two numerical examples and vehicle vibration control application are discussed to substantiate the efficacy of the proposed approach.

Discrete-Time Zhang Neural Network for Online Time-Varying Nonlinear Optimization With Application to Manipulator Motion Generation

In this brief, a discrete-time Zhang neural network (DTZNN) model is first proposed, developed, and investigated for online time-varying nonlinear optimization (OTVNO). Then, Newton iteration is shown to be derived from the proposed DTZNN model. In addition, to eliminate the explicit matrix-inversion operation, the quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is introduced, which can effectively approximate the inverse of Hessian matrix. A DTZNN-BFGS model is thus proposed and investigated for OTVNO, which is the combination of the DTZNN model and the quasi-Newton BFGS method. In addition, theoretical analyses show that, with step-size h=1 and/or with zero initial error, the maximal residual error of the DTZNN model has an O(τ(2)) pattern, whereas the maximal residual error of the Newton iteration has an O(τ) pattern, with τ denoting the sampling gap. Besides, when h ≠ 1 and h ∈ (0,2) , the maximal steady-state residual error of the DTZNN model has an O(τ(2)) pattern. Finally, an illustrative numerical experiment and an application example to manipulator motion generation are provided and analyzed to substantiate the efficacy of the proposed DTZNN and DTZNN-BFGS models for OTVNO.

A Neurodynamic Optimization Method for Recovery of Compressive Sensed Signals With Globally Converged Solution Approximating to l0 Minimization

Finding the optimal solution to the constrained l0 -norm minimization problems in the recovery of compressive sensed signals is an NP-hard problem and it usually requires intractable combinatorial searching operations for getting the global optimal solution, unless using other objective functions (e.g., the l1 norm or lp norm) for approximate solutions or using greedy search methods for locally optimal solutions (e.g., the orthogonal matching pursuit type algorithms). In this paper, a neurodynamic optimization method is proposed to solve the l0 -norm minimization problems for obtaining the global optimum using a recurrent neural network (RNN) model. For the RNN model, a group of modified Gaussian functions are constructed and their sum is taken as the objective function for approximating the l0 norm and for optimization. The constructed objective function sets up a convexity condition under which the neurodynamic system is guaranteed to obtain the globally convergent optimal solution. An adaptive adjustment scheme is developed for improving the performance of the optimization algorithm further. Extensive experiments are conducted to test the proposed approach in this paper and the output results validate the effectiveness of the new method.

A two-layer recurrent neural network for nonsmooth convex optimization problems

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.

Nonlinear model predictive control based on collective neurodynamic optimization

In general, nonlinear model predictive control (NMPC) entails solving a sequential global optimization problem with a nonconvex cost function or constraints. This paper presents a novel collective neurodynamic optimization approach to NMPC without linearization. Utilizing a group of recurrent neural networks (RNNs), the proposed collective neurodynamic optimization approach searches for optimal solutions to global optimization problems by emulating brainstorming. Each RNN is guaranteed to converge to a candidate solution by performing constrained local search. By exchanging information and iteratively improving the starting and restarting points of each RNN using the information of local and global best known solutions in a framework of particle swarm optimization, the group of RNNs is able to reach global optimal solutions to global optimization problems. The essence of the proposed collective neurodynamic optimization approach lies in the integration of capabilities of global search and precise local search. The simulation results of many cases are discussed to substantiate the effectiveness and the characteristics of the proposed approach.

Passivity and passification of memristor-based recurrent neural networks with time-varying delays

This paper presents new theoretical results on the passivity and passification of a class of memristor-based recurrent neural networks (MRNNs) with time-varying delays. The casual assumptions on the boundedness and Lipschitz continuity of neuronal activation functions are relaxed. By constructing appropriate Lyapunov-Krasovskii functionals and using the characteristic function technique, passivity conditions are cast in the form of linear matrix inequalities (LMIs), which can be checked numerically using an LMI toolbox. Based on these conditions, two procedures for designing passification controllers are proposed, which guarantee that MRNNs with time-varying delays are passive. Finally, two illustrative examples are presented to show the characteristics of the main results in detail.

Robust model predictive control of nonlinear systems with unmodeled dynamics and bounded uncertainties based on neural networks

This paper presents a neural network approach to robust model predictive control (MPC) for constrained discrete-time nonlinear systems with unmodeled dynamics affected by bounded uncertainties. The exact nonlinear model of underlying process is not precisely known, but a partially known nominal model is available. This partially known nonlinear model is first decomposed to an affine term plus an unknown high-order term via Jacobian linearization. The linearization residue combined with unmodeled dynamics is then modeled using an extreme learning machine via supervised learning. The minimax methodology is exploited to deal with bounded uncertainties. The minimax optimization problem is reformulated as a convex minimization problem and is iteratively solved by a two-layer recurrent neural network. The proposed neurodynamic approach to nonlinear MPC improves the computational efficiency and sheds a light for real-time implementability of MPC technology. Simulation results are provided to substantiate the effectiveness and characteristics of the proposed approach.

Robust pole assignment for synthesizing feedback control systems using recurrent neural networks

This paper presents a neurodynamic optimization approach to robust pole assignment for synthesizing linear control systems via state and output feedback. The problem is formulated as a pseudoconvex optimization problem with robustness measure: i.e., the spectral condition number as the objective function and linear matrix equality constraints for exact pole assignment. Two coupled recurrent neural networks are applied for solving the formulated problem in real time. In contrast to existing approaches, the exponential convergence of the proposed neurodynamics to global optimal solutions can be guaranteed even with lower model complexity in terms of the number of variables. Simulation results of the proposed neurodynamic approach for 11 benchmark problems are reported to demonstrate its superiority.