A nonlinearly-activated neurodynamic model and its finite-time solution to equality-constrained quadratic optimization with nonstationary coefficients☆
Graphical abstract
Section snippets
Introductionintroduction
Quadratic optimization arises in diverse fields of science and engineering including communication processing [1], image processing [2], nonlinear control [3], motion planning and obstacle avoidance in robotics [4], [5], etc. In addition, various realistic issues can be addressed by turning initial problems into quadratic optimization problems subject to equality constraints. For instance, the least square problem with linear-equality constraints can be regarded as a basic analytical form that
Problem formulation
Let us discuss the following nonstationary quadratic optimization which is subject to the nonstationary equality constraint f(x(t), t) = 0 ∈ Rm:where variable x(t) ∈ Rn is unknown at time instant t ∈ [0, + ∞) and needs to be found [with xT(t) denoting the transpose of x(t)]. In nonstationary quadratic optimization depicted in Eqs. (1), (2), Hessian matrix Q(t) ∈ Rn×n and coefficients p(t) ∈ Rn are smoothly nonstationary. Besides, f(·) denotes a mapping
Related work: GD and ZD models
In this section, for completeness of this paper and for comparative purposes, we present the two of the most relevant works: the GD and ZD models. The GD method is widely used to solve constant problems and the ZD method is recently proposed to solve nonstationary problems. Next, such two methods are developed and exploited to solve the equality-constrained quadratic optimization with nonstationary coefficients.
Nonlinearly-activated neurodynamic model
Because of the in-depth study on neural dynamics, we found that the convergence rate of neural-dynamic models can be thoroughly improved by an elaborate design of the activation function Ψ(·). In addition, taking advantage of the nonlinearity, a properly-designed nonlinear activation function often outperforms the linear one in convergence rate. Therefore, in this section, we aim at developing a nonlinear activation function, which can endow ZD model (15) with a finite-time convergence for
Comparison verification
In the above section, FTCZD model (18) is proposed for solving nonstationary quadratic optimization (4), (5) by adding a specially-constructed nonlinear activation function. In addition to detailed design process, the excellent finite-time convergence performance is analyzed in details. Besides, two of the most relevant works (i.e., the GD model and ZD model) are presented for comparative purposes. In this section, one illustrative example is provided for substantiating the efficacy and
Conclusions
In this paper, by adopting a specially-constructed nonlinear activation function, a finite-time convergent neurodynamic model (i.e., the FTCZD model) has been proposed and studied for real-time solution of nonstationary quadratic optimization problems. The finite-time convergence performance of the proposed model has been analyzed and presented with the convergence upper bound also estimated. We not only have made a comparison among the GD model, the ZD model and the proposed FTCZD model
Acknowledgments
The authors would like to thank the editors and anonymous reviewers for their valuable suggestions and constructive comments which have really helped the authors improve very much the presentation and quality of this paper.
References (28)
- et al.
Repetitive motion planning of PA10 robot arm subject to joint physical limits and using LVI-based primal-dual neural network
Mechatronics
(2008) - et al.
Li-function activated ZNN with finite-time convergence applied to redundant-manipulator kinematic control via time-varying Jacobian matrix pseudoinversion
Appl. Soft Comput.
(2014) - et al.
Zhang neural network for online solution of time-varying convex quadratic program subject to time-varying linear-equality constraints
Phys. Lett. A
(2009) - et al.
Particle swarm optimization of interval type-2 fuzzy systems for FPGA applications
Appl. Soft Comput.
(2013) - et al.
A modified fuzzy min–max neural network for data clustering and its application to power quality monitoring
Appl. Soft Comput.
(2015) - et al.
A nonlinear model to generate the winner-take-all competition
Commun. Nonlinear Sci. Numer. Simul.
(2013) - et al.
Decentralized kinematic control of a class of collaborative redundant manipulators via recurrent neural networks
Neurocomputing
(2012) - et al.
Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations
Physics D
(2006) - et al.
Different Zhang functions resulting in different ZNN models demonstrated via time-varying linear matrix-vector inequalities solving
Neurocomputing
(2013) - et al.
Finite-time solution to nonlinear equation using recurrent neural dynamics with a specially-constructed activation function
Neurocomputing
(2015)
A finite-time convergent neural dynamics for online solution of time-varying linear complex matrix equation
Neurocomputing
A neural network for constrained optimization with application to CDMA communication systems
IEEE Trans. Circuits Syst. II: Exp. Briefs
A nonlinear image reconstruction technique for ECT using a combined neural network approach
Meas. Sci. Technol.
Constrained nonlinear control allocation with singularity avoidance using sequential quadratic programming
IEEE Trans. Control Syst. Technol.
Cited by (74)
Discrete gradient-zeroing neural network algorithms for handling future quadratic program as well as robot arm via ten-instant formula
2023, Journal of the Franklin InstituteNew zeroing neural network with finite-time convergence for dynamic complex-value linear equation and its applications
2022, Chaos, Solitons and FractalsPose control of constrained redundant arm using recurrent neural networks and one-iteration computing algorithm
2021, Applied Soft ComputingCitation Excerpt :By investigating the kinematics of redundant arm, the position formulation and orientation formulation are presented, which are straightforward and effective. Then, by applying zeroing neural network (ZNN, also termed Zhang neural network) method [23–28] and considering joint angle constraints as well as joint angular velocity constraints, a pose control scheme is proposed for redundant arm, which is presented as an optimization problem [29,30]. By using some mathematical knowledge, the pose control scheme is reformulated as a standard quadratic programming (QP) [17] at joint angular velocity level.
Noise-tolerant gradient-oriented neurodynamic model for solving the Sylvester equation
2021, Applied Soft ComputingCitation Excerpt :They transform the coefficient matrix into a trigonometric type and a Hessenberg type by orthogonal similarity transformation, and then solve the transformed equation by the back-substitution method. However, when the approaches are applied to large size matrix equations, they show noticeable deficiencies [26,27]. Specifically, the direct methods may generate many non-zero elements when solving large sparse linear equations, resulting in a significant increase in computation.
Modified Newton integration neural algorithm for solving the multi-linear M-tensor equation
2020, Applied Soft Computing JournalDiscrete ZNN models of Adams-Bashforth (AB) type solving various future problems with motion control of mobile manipulator
2020, NeurocomputingCitation Excerpt :Therefore, it is contributive to develop efficacious models particularly for time-varying LS. Not only that, developing effective models for solving various time-varying problems is significative, e.g., time-varying quadratic programming [11–15]. Neural networks have become an important part of scientific research, attracting the attention of mathematics, engineering and so on [1,12,16,17].
- ☆
This work is supported by the National Natural Science Foundation of China (grant nos. 61503152, 61563017, 61561022, 61363073 and 61363033), the Research Foundation of Education Bureau of Hunan Province, China (grant nos. 15B192 and 15C1119) and the Research Foundation of Jishou University, China (grant nos. 2015SYJG034, JDLF2015013 and 15JDX020).