Elsevier

Applied Soft Computing

Volume 40, March 2016, Pages 252-259
Applied Soft Computing

A nonlinearly-activated neurodynamic model and its finite-time solution to equality-constrained quadratic optimization with nonstationary coefficients

https://doi.org/10.1016/j.asoc.2015.11.023Get rights and content

Highlights

  • For the first time, a novel finite-time convergent neurodynamic model is proposed.

  • The conventional GD model and the recently-proposed ZD model are developed.

  • Our proposed neurodynamic model can outperform the existing neural dynamical models.

Abstract

The recently-proposed Zhang dynamics (ZD) has been proven to achieve success for solving the linear-equality constrained time-varying quadratic program ideally when time goes to infinity. The convergence performance is a significant improvement, as compared to the gradient-based dynamics (GD) that cannot make the error converge to zero even after infinitely long time. However, this ZD model with the suggested activation functions cannot reach the theoretical time-varying solution in finite time, which may limit its applications in real-time calculation. Therefore, a nonlinearly-activated neurodynamic model is proposed and studied in this paper for real-time solution of the equality-constrained quadratic optimization with nonstationary coefficients. Compared with existing neurodynamic models (specifically the GD model and the ZD model) for optimization, the proposed neurodynamic model possesses the much superior convergence performance (i.e., finite-time convergence). Furthermore, the upper bound of the finite convergence time is derived analytically according to Lyapunov theory. Both theoretical and simulative results verify the efficacy and superior of the nonlinearly-activated neurodynamic model, as compared to these of the GD and ZD models.

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:minimize12xT(t)Q(t)x(t)+pT(t)x(t),subjecttof(x(t),t)=0,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.

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    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).

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