Physics Letters A 373 (2009) 1639–1643

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Physics Letters A

www.elsevier.com/locate/pla

Zhang neural network for online solution of time-varying convex quadraticprogram subject to time-varying linear-equality constraints

Yunong Zhang ∗, Zhan Li

School of Information Science and Technology, Sun Yat-sen University, Guangzhou 510275, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 November 2008Received in revised form 18 December 2008Accepted 5 March 2009Available online 14 March 2009Communicated by A.R. Bishop

PACS:05.10.-a

MSC:41A2552A4192B20

Keywords:Recurrent neural networksTime-varyingQuadratic programmingGlobal convergenceGradient-based neural network (GNN)

In this Letter, by following Zhang et al.’s method, a recurrent neural network (termed as Zhangneural network, ZNN) is developed and analyzed for solving online the time-varying convex quadratic-programming problem subject to time-varying linear-equality constraints. Different from conventionalgradient-based neural networks (GNN), such a ZNN model makes full use of the time-derivativeinformation of time-varying coefficient. The resultant ZNN model is theoretically proved to have globalexponential convergence to the time-varying theoretical optimal solution of the investigated time-varyingconvex quadratic program. Computer-simulation results further substantiate the effectiveness, efficiencyand novelty of such ZNN model and method.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

The problems of quadratic optimization constrained with lin-ear equalities are widely encountered in various applications; e.g.,optimal controller design [1], power-scheduling [2], robot-arm mo-tion planning [3], and digital signal processing [4]. In addition,nonlinear optimization problems could usually be approximatedby a second-order quadratic system and then solved by a numeri-cal quadratic-programming technique sequentially [5,6]. It is worthnoting that the minimal arithmetic operations of a quadratic-programming (QP) algorithm are normally proportional to the cubeof its Hessian matrix’s dimension, and consequently such numer-ical algorithms may be not efficient enough for large-scale onlineapplications [4].

Being another important type of solution to optimization prob-lems, many parallel-processing computational methods have beenproposed, developed, analyzed, and implemented based on specificarchitectures, e.g., the neural-dynamic and analog solvers in [7–10].

* Corresponding author. Tel.: +86 20 84113572; fax: +86 20 84113572.E-mail addresses: zhynong@mail.sysu.edu.cn, ynzhang@ieee.org (Y. Zhang),

lizhan@mail2.sysu.edu.cn (Z. Li).

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2009.03.011

Such a neural-dynamic approach is now regarded as a powerfulalternative to real-time computation and optimization owing toits parallel-processing distributed nature and convenience of hard-ware implementation [11–18].

However, many numerical methods and neural-dynamic ap-proaches have been designed intrinsically for solving quadratic-programs subject to constant coefficients and/or constraints only.Such computational algorithms may be efficient for some appli-cations, however, for time-varying applications requiring a fasterconvergence to exact solutions, superior approaches might beneeded or desired. Different from the conventional gradient orgradient-based methods (GNN) [9,19,20], in this Letter a new re-current neural network model is proposed, analyzed and simulatedfor the online solution of time-varying quadratic program sub-ject to time-varying linear-equality constraints by following Zhanget al.’s design method [15–18]. For presentation convenience, thename ‘Zhang neural network (ZNN)’ is used in this Letter, whichmight help readers understand better its difference from the well-known gradient-based approach and its related neural-networkmodels.

To the best of our knowledge, there is almost no other liter-ature dealing with such specific time-varying quadratic-programssubject to time-varying linear constraints at present stage, and

1640 Y. Zhang, Z. Li / Physics Letters A 373 (2009) 1639–1643

the main contributions of the Letter may lie in the followingfacts.

(1) The online solution of time-varying quadratic programs sub-ject to time-varying constraints is investigated in this Letter,rather than conventionally-investigated static quadratic pro-grams subject to static constraints [19,21,22].

(2) In this Letter, to solve the time-varying quadratic program, animplicit-dynamic ZNN model is designed based on a vector-valued indefinite error function, rather than usually-usedscalar-valued nonnegative energy functions associated usuallywith GNN (or Hopfield-type networks).

(3) A general framework of the ZNN model is proposed for solvingthe quadratic program with time-varying coefficients, which isguaranteed to have global convergence to its theoretical time-varying optimal solution. By using linear and power-sigmoidactivation functions, the ZNN model could then have globalexponential convergence and superior convergence to the the-oretical time-varying optimal solution, respectively.

(4) An illustrative verification-and-comparison example is pre-sented, where the ZNN and GNN models are both used tosolve the time-varying quadratic program. The novelty, differ-ence and efficacy of the proposed ZNN model are thus shownevidently.

2. Problem formulation and preliminaries

Consider the following time-varying convex quadratic program-ming problem which is subject to the time-varying linear-equalityconstraints A(t)x(t) = b(t):

minimize xT (t)P (t)x(t)/2 + qT (t)x(t),

subject to A(t)x(t) = b(t), (1)

where vector x(t) ∈ Rn at time instant t ∈ [0,+∞) is unknownand to be solved for in real time. In time-varying quadratic pro-gram (1), Hessian matrix P (t) ∈ Rn×n , coefficient vector q(t) ∈ Rn ,coefficient matrix A(t) ∈ Rm×n (being of full row rank), and coeffi-cient vector b(t) ∈ Rm are smoothly time-varying. In addition, thecoefficient matrices and vectors, together with their time deriva-tives, are assumed to be known or could be estimated accurately.To guarantee the solution uniqueness, such time-varying quadraticprogram (1) should be strictly convex [21,22] with P (t) ∈ Rn×n

positive-definite at any time instant t ∈ [0,+∞) throughout thisLetter.

Facing time-varying quadratic program (1) and based on thepreliminary results on equality-constrained optimization problems[21–23], we have its related Lagrangian:

L(x(t), λ(t), t

) = xT (t)P (t)x(t)/2 + qT (t)x(t)

+ λT (t)(

A(t)x(t) − b(t)),

where λ(t) ∈ Rm denotes the Lagrange-multiplier vector.As we may recognize or know, solving the time-varying quad-

ratic program (1) could be done by zeroing the equations below atany time instant t ∈ [0,+∞):{

∂L(x(t),λ(t),t)∂x(t) = P (t)x(t) + q(t) + AT (t)λ(t) = 0,

∂L(x(t),λ(t),t)∂λ(t) = A(t)x(t) − b(t) = 0.

The above equations could be further written as

W (t)y(t) = u(t), (2)

where

W (t) :=[

P (t) AT (t)A(t) 0m×m

]∈ R(n+m)×(n+m),

y(t) :=[

x(t)

λ(t)

]∈ Rn+m, u(t) :=

[−q(t)

b(t)

]∈ Rn+m.

Moreover, for the purposes of better understanding and compari-son, we know that the time-varying theoretical solution could bewritten as

y∗(t) = [x∗T

(t), λ∗T(t)

]T := W −1(t)u(t) ∈ Rn+m.

To demonstrate the significance of the interesting problem (1)as well as for better visual effects and readability, we can take thefollowing specific time-varying quadratic program as an example:

minimize((sin t)/4 + 1

)x2

1(t) + ((cos t)/4 + 1

)x2

2(t)

+ cos tx1(t)x2(t) + sin 3tx1(t) + cos 3tx2(t),

subject to sin 4tx1(t) + cos 4tx2(t) = cos 2t. (3)

Fig. 1 shows the three-dimension snapshots at different time in-stants (i.e., t = 0.00, 2.15, 5.80 and 6.35 s), where horizontal axescorrespond to x1(t) ∈ R and x2(t) ∈ R . From the figure, we cansee evidently that the linear-constraint plane and the objective-function surface are both “moving” with time t . In other words,the problem that we are going to solve [namely, (1)] is a quitechallenging problem in the sense that the optimal solution is also“moving” with time t due to the “moving” effects of the objective-function surface and linear-constraint plane.

3. Neural network solvers and comparison

In the literature, conventional gradient-based neural networksand computational algorithms have been developed to computealgebraic and optimization problems with constant coefficients[9,19,21,22]. However, when applied to the time-varying situation,a much faster convergence rate is required for these gradient-basedapproaches as compared to the variational rate of coefficient ma-trices and vectors. This may thus impose very stringent restrictionson hardware realization and/or sacrifice the solution precision verymuch (see Fig. 1 of [24]). Different from the above gradient-basedapproaches, the following Zhang neural network is proposed inthis Letter for finding the optimal solution of the time-varyingquadratic program (1).

3.1. Zhang neural network

To monitor the solving process of time-varying quadratic pro-gram (1) via time-varying linear system (2), we could firstly definethe following vector-valued error function (rather than the scalar-valued nonnegative energy functions used in gradient-based neuralapproaches):

e(t) = W (t)y(t) − u(t) ∈ Rn+m, (4)

of which each element could be positive, negative, and lower-unbounded. Then, to make each element ei(t) of error vectore(t) ∈ Rn+m converge to zero, the following ZNN design formulacan be adopted [15–18,24]:

de(t)

dt= −γΦ

(e(t)

), (5)

where design-parameter γ > 0, being a set of reciprocals ofcapacitance-parameters, should be set as large as hardware per-mits [25] or set appropriately for simulative/experimental pur-poses. Φ(·) : Rn+m → Rn+m denotes an activation-function pro-cessing-array. In addition, each scalar-valued processing-unit φ(·)of array Φ(·) should be a monotonically-increasing odd activationfunction.

Y. Zhang, Z. Li / Physics Letters A 373 (2009) 1639–1643 1641

Fig. 1. “Moving” quadratic function, “Moving” linear constraint and “Moving” optimal solution of time-varying quadratic program (1).

Expanding ZNN design formula (5) leads to the following ZNNmodel depicted in an implicit-dynamic equation:

W (t) y(t) = −W (t)y(t) − γΦ(W (t)y(t) − u(t)

) + u(t), (6)

where state vector y(t) ∈ Rn+m , starting from an initial condi-tion y(0) ∈ Rn+m , corresponds to the theoretical solution of (2), ofwhich the first n elements constitute the neural-network solutioncorresponding to the optimal solution of time-varying quadraticprogram (1).

3.2. Comparison with gradient neural network

For comparison, it is worth pointing out here that we candevelop a gradient-based neural network to solve online thequadratic program (1). However, similar to almost all numericalalgorithms and neural-dynamic schemes mentioned before, thegradient neural networks are designed intrinsically for problemswith constant coefficient matrices and/or vectors [9,19,20]. Nowwe show the GNN design procedure as the following.

(1) Firstly, a scalar-valued norm-based energy function, such as‖W y − u‖2

2/2 with ‖ · ‖2 denoting the two norm of a vec-tor, is constructed such that its minimum point is the solutionof linear system W y = u.

(2) Secondly, an algorithm is designed to evolve along a descentdirection of this energy function until the minimum point isreached. The typical descent direction is the negative of thegradient of energy function ‖W y − u‖2

2/2, i.e.,

−∂‖W y − u‖22/2

∂ y= −W T (W y − u).

(3) Thirdly, by using the above negative gradient to construct andapply the neural network to the time-varying situation, wecould have a linear GNN model solving (1),

y(t) = −γ W T (t)(W (t)y(t) − u(t)

),

and a generalized nonlinear GNN model,

y(t) = −γ W T (t)Φ(W (t)y(t) − u(t)

). (7)

1642 Y. Zhang, Z. Li / Physics Letters A 373 (2009) 1639–1643

Fig. 2. Online solution of time-varying quadratic program (3) by ZNN and GNN models with design parameter γ = 1, where dotted-circle and dotted-plus curves correspondto the time-varying neural-network solution y(t) synthesized respectively by ZNN and GNN models, with dotted curves corresponding to the time-varying theoretical solutiony∗(t).

The main differences and novelties between ZNN (6) and GNN(7) may lie in the following facts.

(1) ZNN model (6) is designed based on the elimination of everyelement of the vector-valued error function e(t) = W (t)y(t) −u(t). In contrast, GNN model (7) is designed based on theelimination of the scalar-valued norm-based energy function‖W y − u‖2

2.(2) ZNN model (6) is depicted in an implicit dynamics, i.e.,

W (t) y(t) = · · · , which coincides well with systems in na-ture and in practice (e.g., in analogue electronic circuits andmechanical systems owing to Kirchhoff ’s and Newton’s laws,respectively [7,16,18,26]). In contrast, GNN model (7) is de-picted in an explicit dynamics, i.e., y(t) = · · · .

(3) ZNN model (6) could systematically and methodologically ex-ploit the time-derivative information of coefficient matricesand vectors during its real-time solving process. In contrast,GNN model (7) has not exploited such information, thus lesseffective on time-varying problems solving.

4. Convergence analysis and results

In this section, three theorems about the convergence proper-ties of ZNN model (6) are established for online solution of time-varying quadratic program (1). The analysis includes the situationsof using linear or power-sigmoid activation functions. Due to space

limitation, the proofs of the following theorems are omitted butcan be generalized from [16,18] by taking into account Lyapunovfunction candidate v = eT (t)e(t)/2 and using Lyapunov theory[27,28].

Theorem 1. Consider smoothly time-varying strictly-convex quadraticprogram (1). If a monotonically-increasing odd activation-function ar-ray Φ(·) is used, then state vector y(t) of ZNN (6), starting from anyinitial state y(0) ∈ Rn+m, could globally converge to the unique the-oretical solution y∗(t) = [x∗T (t), λ∗T (t)]T of time-varying linear sys-tem (2). In addition, the first n elements of solution y∗(t) constitute thetime-varying optimal solution x∗T (t) to the time-varying quadratic pro-gram (1).

Theorem 2. In addition to Theorem 1, if the linear activation functionφ(ei(t)) = ei(t) is used, then state vector y(t) of ZNN (6) could glob-ally exponentially converge to the unique theoretical solution y∗(t) =[x∗T (t), λ∗T (t)]T of time-varying linear system (2), with x∗(t) being thetime-varying optimal solution to time-varying problem (1).

Theorem 3. In addition to Theorems 1 and 2, if we use the power-sigmoid activation function

φ(ei(t)) ={

e2r−1i (t), |ei(t)| � 1,

1+exp(−ζ ) 1−exp(−ζei(t)) , |ei(t)| � 1

1−exp(−ζ ) 1+exp(−ζei(t))

Y. Zhang, Z. Li / Physics Letters A 373 (2009) 1639–1643 1643

with suitable design parameters ζ � 2 and r � 2 (being an integer), thenstate vector y(t) of ZNN model (6) is superiorly globally convergent to thetheoretical time-varying solution y∗(t) = [x∗T (t), λ∗T (t)]T with x∗(t)being the optimal solution to time-varying quadratic program (1), whichis compared with the situation of using linear activation functions aspresented in Theorem 2.

5. Illustrative example

In this section we show an illustrative computer-simulationexample so as to demonstrate the characteristics of the neural-network convergence. For illustration and comparison purposes,both ZNN model (6) and GNN model (7) are employed to solveonline such a time-varying quadratic program (3), which exploitpower-sigmoid activation functions with design parameters r = 2and ζ = 4.

As seen from Fig. 2, starting form randomly-generated initialstate y(0) = [x1(0), x2(0), λ1(0)]T ∈ [−1,1]3, state-vector y(t) ofZNN model (6) could always globally converge to the theoreticaltime-varying solution y∗(t) = [x∗

1(t), x∗2(t), λ

∗1(t)]T exactly, of which

the x∗1(t) and x∗

2(t) are the time-varying optimal solution to time-varying quadratic program (3). In contrast, as shown in the samefigure (i.e., Fig. 2), state-vector y(t) of GNN model (7) do not fitwell with the theoretical solution y∗(t) with quite large computa-tional errors.

Moreover, we could also monitor the residual error ‖W (t)y(t)−u(t)‖2 during the problem-solving process as of ZNN and GNNmodels. The ZNN steady-state residual-error limt→+∞ ‖W (t)y(t)−u(t)‖2 is around 2.842700 × 10−13 and 2.661423 × 10−13, whichcorrespond to the use of design parameter γ = 1 and 10, respec-tively. In contrast, by using GNN (7) to solve the time-varyingquadratic program (3) under the same simulation conditions, itssteady-state residual-error limt→+∞ ‖W (t)y(t) − u(t)‖2 is about1.461293 and 0.814059 (corresponding to the use of design param-eter γ = 1 and 10, respectively), much larger than the error-valuessynthesized by ZNN model (6).

6. Concluding remarks

In this Letter, a Zhang neural network model is proposed, inves-tigated and applied to time-varying quadratic-program solving. Dif-ferent from conventional gradient-based neural-network methodsand models, the Zhang neural network could methodologically uti-lize the time-derivative information of time-varying problems andthus achieve global exponential convergence to the time-varying

optimal solution. Computer-simulation results via power-sigmoidactivation functions have further substantiated the effectiveness,efficiency and superiority of the ZNN model for online solutionof time-varying quadratic programming problems subject to time-varying linear constraints.

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