Applied Mathematics and Computation 206 (2008) 186–192

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

A stable differential equation approach for inequality constrainedoptimization problems q

Li JinSchool of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan, Zhejiang 316004, PR China

a r t i c l e i n f o

Keywords:Inequality constrained optimizationDifferential equationAsymptotical stabilityQuadratic convergenceModified barrier function

0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.09.007

q This material is based upon work funded by ZheE-mail address: jinli_70@yahoo.com.cn

a b s t r a c t

This paper presents a first-order derivatives based and a second-order derivatives baseddifferential equation systems for inequality constrained optimization problems by usingthe modified barrier function. Under the suitable conditions, we prove the asymptotic sta-bility of the two differential systems and local convergence properties of their Euler dis-crete schemes, including the locally quadratic convergence rate of the discrete algorithmfor second-order derivatives based differential equation system. Numerical tests are pre-sented that confirm the robustness and efficiency of the approach.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

The first differential equation method for solving equality constrained optimization problems was proposed by Arrow andHurwicz [1]. Fiacco and McCormick [2] used differential equation systems for studying optimality conditions for nonlinearprogramming problems. A systematic study on differential equation methods for solving nonlinear optimization startedfrom the work of Evtushenko and Zhadan in early 1970s, see for instances Yamadhita [3], Pan [4], Evtushenko [5,6] andEvtushenko and Zhadan [7–9]. The main idea of this type of methods is to construct a system of differential equations suchthat the equilibrium points of this system coincide with the solutions (or KKT points) to the constrained optimization prob-lem. Along this line, Zhang [10] and Zhang et al. [11] studied a modified Evtushenko–Zhadan systems for solving equalityconstrained optimization problems, Jin et al. [12] proposed differential equation systems for inequality constrainedoptimization.

This paper constructs a simple differential system for solving inequality constrained optimization problems. In Section 2,the modified barrier function, F(x,y, t), is proposed, based on which the system of differential equations is constructed. It isproved that KKT points of an inequality constrained optimization problem is its asymptotically stable equilibrium points.Euler discrete schemes with constant stepsizes for the differential system is presented and the convergence theorem is dem-onstrated. In Section 3, the second differential system is generated by using the second-order derivatives of problem func-tions. We prove its asymptotic stability and local convergence property of its Euler discrete schemes, including the locallyquadratic convergence rate of the discrete algorithm with Armijo line searches. Finally, the algorithm with Armijo linesearches and the Runge–Kutta algorithm for the differential system are employed to solve several numerical examples,the numerical results show that the Runge–Kutta method has better stability and higher precision than the Armijo stepsizemethod.

. All rights reserved.

jiang Provincial Natural Science Foundation of China under Grant No. Y6080388.

L. Jin / Applied Mathematics and Computation 206 (2008) 186–192 187

2. A differential system

Consider the inequality constrained optimization problems

min f ðxÞ;s:t: giðxÞP 0; i ¼ 1; . . . ;m;

ð2:1Þ

where f ; gi : Rn ! R; i ¼ 1; . . . ;m. Here each function is twice continuously differentiable. Let Lðx;lÞ ¼ f ðxÞ þPm

i¼1ligiðxÞ bethe (classical) Lagrangian for problem (2.1) and IðxÞ ¼ fijgiðxÞ ¼ 0; i ¼ 1; . . . ;mg denote the active set of indices at x. Let x� bea local minimizer to problem (2.1) and ðx�;l�Þ be the corresponding KKT point, which satisfies the following conditions:

rxLðx�;l�Þ ¼ rf ðx�Þ �Xm

i¼1

l�irgiðx�Þ ¼ 0;

l� P 0; l�i giðx�Þ ¼ 0; giðx�ÞP 0; i ¼ 1; . . . ;m:

ð2:2Þ

Furthermore, let the Jacobian uniqueness conditions, proposed in [2], hold at ðx�;l�Þ

(1) The multipliers l� > 0; i 2 Iðx�Þ.(2) The gradients rgiðx�Þ; i 2 Iðx�Þ are linearly independent.(3) yTr2

xxLðx�;l�Þy > 0 8 0–y 2 fyjrgiðx�ÞTy ¼ 0; i 2 Iðx�Þg.

To find a solution of problem (2.1), we propose a nonlinear Lagrangian for problem (2.1), in the following form:

Fðx; y; tÞ ¼ f ðxÞ � tXm

i¼1

y2i lgðt�1giðxÞ þ 1Þ ðt > 0Þ;

Let zT ¼ ðxT; yTÞ, by differentiating F(x,y,t) with respect to z, we construct the following system of differential equations:

dzdt¼

�rf ðxÞ þPmi¼1

y2irgiðxÞ

t�1giðxÞþ1

�2ty1 lgðt�1g1ðxÞ þ 1Þ...

�2tym lgðt�1gmðxÞ þ 1Þ

2666664

3777775ðt > 0Þ: ð2:3Þ

The following lemma will be used in the proof of the forthcoming theorem.

Lemma 2.1 [13]. Let A be a n� n symmetrical matrix, B be a r � n matrix, U ¼ ½diagli�mi¼1, where l ¼ ðl1; . . . ;lrÞ > 0. If k > 0

is a scalar and

By ¼ 0 implies hAy; yiP khy; yi:

Then there are scalars k0 > 0 and c 2 ð0; kÞ such that, for any k P k0,

hðAþ kBTUBÞx; xiP chx; xi 8x 2 Rn:

Lemma 2.2. Let ðx�;l�Þ be a KKT point of (2.1), the Jacobian uniqueness conditions hold at ðx�;l�Þ. Then there is y� 2 Rm, suchthat ðy�i Þ

2 ¼ l�i ði ¼ 1; . . . ;mÞ, the following conclusions are satisfied:

(i) Fðx�; y�; tÞ ¼ Lðx�;l�Þ ¼ f ðx�Þ.(ii) rxFðx�; y�; tÞ ¼ rxLðx�;l�Þ ¼ 0.

(iii) r2xxFðx�; y�; tÞ ¼ r2

xxLðx�;l�Þ þ t�1rgðx�ÞTDrgðx�Þ,where D ¼ ½diagðy�i Þ

2=ðt�1giðx�Þ þ 1Þ2�mi¼1;rgðx�Þ ¼ ½rg1ðx�Þ; . . . ;rgrðx�Þ�

T.(iv) There is t0 > 0 and c > 0, for any t 2 ð0; t0�, such that

hr2xxFðx�; y�; tÞw;wiP chw;wi 8w 2 Rn:

Proof. Without loss of generality, we assume that Iðx�Þ ¼ f1; . . . ; rg, where r 6 m. It follows from the KKT conditions that

Fðx�; y�; tÞ ¼ f ðx�Þ � tXm

i¼1

ðy�i Þ2 lgðt�1giðx�Þ þ 1Þ ¼ f ðx�Þ � t

Xr

i¼1

ðy�i Þ2 lgðt�1giðx�Þ þ 1Þ ¼ f ðx�Þ;

where ðy�i Þ2 ¼ l�i ði ¼ 1; . . . ;mÞ. Furthermore, Fðx; y; tÞ is differential at x, then we have

rxFðx�; y�; tÞ ¼ rf ðx�Þ �Xm

i¼1

ðy�i Þ2rgiðx�Þ

t�1giðx�Þ þ 1¼ rf ðx�Þ �

Xm

i¼1

l�irgiðx�Þ ¼ rxLðx�;l�Þ ¼ 0:

188 L. Jin / Applied Mathematics and Computation 206 (2008) 186–192

By calculating, we get

r2xxFðx�; y�; tÞ ¼ r2f ðx�Þ �

Xm

i¼1

ðy�i Þ2r2giðx�Þ

t�1giðx�Þ þ 1þXm

i¼1

ðy�i Þ2rgiðx�Þrgiðx�Þ

T

ðt�1giðx�Þ þ 1Þ2¼ r2

xxLðx�;l�Þ þ t�1rgðx�ÞTDrgðx�Þ;

where D ¼ ½diagl�i �mi¼1. Let A ¼ r2

xxLðx�;l�Þ; B ¼ rgðx�Þ and U ¼ D in Lemma 2.1, using the Jacobian uniqueness conditions,we obtain (iv). The proof is completed. h

Lemma 2.3. Let the Jacobian uniqueness conditions hold at ðx�;l�Þ. Then ðx�; y�Þ is an equilibrium point of system (2.3) if and onlyif ðx�;l�Þ be a KKT point of (2.1), where ðy�i Þ

2 ¼ l�i ; ði ¼ 1; . . . ;mÞ.

Proof. Let ðx�;l�Þ be a Kuhn–Tucker point of (2.1), by introducing ðy�i Þ2 ¼ l�i ði ¼ 1; . . . ;mÞ, we have from Lemma 2.2 that

rxFðx�; y�; tÞ ¼ rxLðx�;l�Þ ¼ 0;

which implies that ðx�; y�Þ is an equilibrium point of (2.2).On the contrary, let ðx�; y�Þ be an equilibrium point of (2.2). Existing l�i ¼ ðy�i Þ

2ði ¼ 1; . . . ;mÞ, we have from Lemma 2.2 andthe Jacobian uniqueness conditions that

rxLðx�;l�Þ ¼ 0; l� P 0; l�i giðx�Þ ¼ 0; giðx�Þ 6 0; i ¼ 1; . . . ;m

implying that ðx�;l�Þ be a KKT point of (2.1). h

Theorem 2.1. Let ðx�;l�Þ be a KKT point of (2.1), the Jacobian uniqueness conditions hold at ðx�;l�Þ. Then the system (2.3) witht > 0 is asymptotically stable at ðx�; y�Þ, where ðy�i Þ

2 ¼ l�i ði ¼ 1; . . . ;mÞ.

Proof. Without loss of generality, we assume that Iðx�Þ ¼ f1; . . . ; rgwith r 6 m. Linearizing system (2.3) in the neighborhoodof z�T ¼ ðx�T; y�TÞ, we obtain

dzdt¼ �Qðz� z�Þ; ð2:4Þ

where

Q ¼Q 1 00 Q 2

� �ð2:5Þ

and Q 1 2 RðnþrÞ�ðnþrÞ and Q2 2 Rðm�rÞ�ðm�rÞ are given by

Q 1 ¼

r2xxFðx�; y�; tÞ �2y�1rg1ðx�Þ; . . . ;�2y�rrgrðx�Þ

2y�1rg1ðx�ÞT

..

.0

2y�rrgrðx�ÞT

2666664

3777775

ð2:6Þ

and

Q2 ¼ 2t½diag lgðt�1giðxÞ þ 1Þ�mi¼rþ1:

The stability of system (2.3) is determined by the properties of the roots of the characteristic equation

detðQ � kInþmÞ ¼ 0; ð2:7Þ

which is equivalent to the following two equations:

jQ 1 � kInþrj ¼ 0; jQ2 � kIm�rj ¼ 0:

The solutions of the second equation are explicitly expressed as

ki ¼ 2t lgðt�1giðxÞ þ 1Þ; nþm� r þ 1 6 i 6 nþm

and from the assumptions we have

�k ¼ minnþm�rþ16i6nþm

ki > 0: ð2:8Þ

Let y denote the conjugate vector of a complex vector y, ReðbÞ denote the real part of the complex number b. Let a be aneigenvalue of Q1 and ðz1; z2ÞT 2 Rn � Rr the corresponding nonzero eigenvector of the matrix Q1. Then

Re ½z1; z2�Q 1z1

z2

� �� �¼ Re a½z1; z2�

z1

z2

� �� �¼ ReðaÞðjz1j2 þ jz2j2Þ: ð2:9Þ

It follows from the definition of Q1 that

ReðaÞðjz1j2 þ jz2j2Þ ¼ Refz1Tr2

xxFðx�; y�; tÞz1g: ð2:10Þ

L. Jin / Applied Mathematics and Computation 206 (2008) 186–192 189

In view of Lemma 2.2, the matrix r2xxFðx�; y�; tÞ is positive definite, we obtain for z1–0 that

Refz1Tr2

xxFðx�; y�; tÞz1g > 0: ð2:11Þ

If z1 ¼ 0, then from the Jacobian uniqueness conditions, yields z2 ¼ 0. This contradicts with ðz1; z2Þ–0. Therefore, by notingthat (2.8) and (2.11), we have that all eigenvalues of �Q have negative real parts. It follows from Lyapunov’s stability the-orem of the first-order approximation that ðx�; y�Þ is a local asymptotically stable equilibrium point of (2.3).

Integrating the system (2.3) by the Euler method, one obtains the iterate process

xkþ1 ¼ xk � hkðrf ðxkÞ �Pmi¼1

ðykiÞ2rgiðxkÞ

t�1giðxkÞþ1Þ ;

ykþ1i ¼ yk

i � 2hkyki ðlg t�1giðxkÞ þ 1Þ; i ¼ 1; . . . ;m;

8><>: ð2:12Þ

where yk ¼ ðyk1; y

k2; . . . ; yk

mÞT and hk is a stepsize. h

The following theorem tells us that the Euler scheme with a constant stepsize is locally convergent.

Theorem 2.2. Let conditions of Theorem 2.1 be satisfied at the point ðx�;l�Þ. Then there exists a �h > 0 such that for any0 < hk <

�h, the iterations defined by (2.12) with t > 0 converge locally to ðx�; y�Þ, where l�i ¼ ðy�i Þ2; i ¼ 1; . . . ;m.

Proof. Let zT ¼ ðxT; yTÞ, since z� is a fixed point of the mapping

FðzÞ ¼ z� t

rf ðxÞ �Pmi¼1

y2irgiðxÞ

t�1giðxÞþ1

2ty1 lgðt�1g1ðxÞ þ 1Þ...

2tym lgðt�1gmðxÞ þ 1Þ

266666664

377777775: ð2:13Þ

The convergence of the iterations (2.12) will be proved if we demonstrate that a �h can be chosen such that the iterationsdefined by

zkþ1 ¼ FðzkÞ

converge to z� whenever z0 is in a neighborhood of z� and 0 < hk <�h. LetrFðzÞ be Jacobian transpose of F(z) and m1; . . . ; mnþm

be the eigenvalues of the matrix rFðz�ÞT with the expression

mj ¼ ð1� tajÞ � iðtbjÞ:

From the proof of Theorem 2.1 one has that aj > 0. The condition jmjj < 1 can be written as

h < 2aj=ða2j þ b2

j Þ; 1 6 j 6 nþm:

Let

�h ¼minf2aj=ða2j þ b2

j Þj1 6 j 6 nþmg;

then the spectral radius ofrFðz�ÞT is strictly less than 1 for h < �h, and the iterations generated by the scheme (2.12) is locallyconvergent to ðx�; y�Þ (see Evtushenko [6]). The proof is completed. h

3. The second differential system

The method in Section 2 can be viewed as a gradient approach, through the system (2.3), for solving problem (2.1). Nowwe discuss the Newton version. The continuous version of Newton method leads to the initial value problem for the follow-ing system of ordinary differential equations:

KðzÞdzdt¼ diag16i6nþmci

�rf ðxÞ þPmi¼1

y2irgiðxÞ

ðt�1giðxÞþ1Þ

�2ty1 lgðt�1g1ðxÞ þ 1Þ

..

.

�2tym lgðt�1gmðxÞ þ 1Þ

2666666664

3777777775; ð3:1Þ

190 L. Jin / Applied Mathematics and Computation 206 (2008) 186–192

where zT ¼ ðxT; yTÞ; ðc1; . . . ; cnþmÞT > 0 is a scaling vector and KðzÞ is the Jacobian matrix of the mapping

/ðzÞ ¼

rf ðxÞ �Pmi¼1

y2irgiðxÞ

ðt�1giðxÞþ1Þ

2ty1 lgðt�1g1ðxÞ þ 1Þ...

2tym lgðt�1gmðxÞ þ 1Þ

266666664

377777775: ð3:2Þ

We can easily get the formula

KðzÞ ¼

r2xxFðx; y; tÞ � 2y1rg1ðxÞ

t�1g1ðxÞþ1� � � � 2ymrgmðxÞ

t�1gmðxÞþ1

2y1rg1ðxÞT

t�1g1ðxÞþ12t lgðt�1g1ðxÞ þ 1Þ

..

. . ..

2ymrgmðxÞT

t�1gmðxÞþ12t lgðt�1gmðxÞ þ 1Þ

26666664

37777775:

Lemma 3.1. Let conditions of Theorem 2.1 be satisfied at the point ðx�;l�Þ. Then, for t > 0, Kðz�Þ is a nonsingular matrix, wherel�i ¼ ðy�i Þ

2ði ¼ 1; . . . ;mÞ and z�T ¼ ðx�T; y�TÞ.

Proof. Let ðx�;l�Þ be a KKT point of (2.1), the Jacobian uniqueness conditions hold at ðx�;l�Þ, and denote Iðx�Þ ¼f1; . . . ; rg; r 6 m. Then we obtain

Kðz�Þ ¼

r2xxFðx�; y�; tÞ �2y�1rg1ðx�Þ � � � �2y�rrgrðx�Þ 0 � � � 0

2y�1rg1ðx�ÞT 0 0

..

. . .. . .

.0

2y�rrgrðx�ÞT 0

0rþ1 0 2t lgðt�1grþ1ðx�Þ þ 1Þ... . .

. . ..

0m 0 2t lgðt�1gmðx�Þ þ 1Þ

266666666666664

377777777777775

:

Since ðx�;l�Þ be a KKT point of (2.1), we have, equivalently, by Lemma 2.2 that the matrix r2xxFðx�; y�; tÞ is positive definite.

Since the Jacobian uniqueness conditions, we have that y�1rg1ðx�Þ; . . . ; y�rrgrðx�Þ are linearly independent. Moreover,giðx�Þ > 0 for all i 2 fr þ 1; . . . ;mg. Hence, we obtain that Kðz�Þ is a nonsingular matrix. h

Theorem 3.1. Let conditions of Theorem 2.1 be satisfied at the point ðx�;l�Þ. Then the system (3.1) with t > 0 is asymptoticallystable at z�, where l�i ¼ ðy�i Þ

2ði ¼ 1; . . . ;mÞ and z�T ¼ ðx�T; y�TÞ.

Proof. From the second-order smoothness of / around z�, we have

/ðzÞ ¼ /ðz�Þ þKðz�ÞdðzÞ þHðdðzÞÞ;

where dðzÞ ¼ z� z�; HðdðzÞÞ ¼ OðkdðzÞk2Þ.Linearizing system (3.1) at the point z�, we obtain

ddðzÞdt¼ �Qðz�ÞdðzÞ; Qðz�Þ ¼ Kðz�Þ�1diag16i6nþmciKðz�Þ:

Matrix Qðz�Þ is similar to matrix diag16i6nþmci; therefore, they have the same eigenvalues ki ¼ ci > 0;1 6 i 6 nþm.According to Lyapunov linearization principle, we have that the equilibrium point z� is asymptotically stable.

Integrating the system (3.1) by the Euler method, one obtains the iterate process

zkþ1 ¼ zk � akdiag16i6nþmciKðzkÞ�1/ðzkÞ: � ð3:3Þ

Theorem 3.2. Let conditions of Theorem 2.1 be satisfied at the point ðx�;l�Þ. Then there exists a �a ¼ 2=max16i6nþmci such that forany 0 < ak < �a, the iterations fzkg defined by (3.3) converge locally to z�.

We adopt the Armijo line search rule to determine step lengths when the Euler method is used. Let EðzÞ ¼ k/ðzÞk2 be themerit function.

Algorithm 3.1

Step 1: Given a 2 ð0;1Þ; ci ¼ 1ð1 6 i 6 nþmÞ;q 2 ð0;1=2Þ; e P 0 and initial point z0 ¼ ðx0; y0Þ; k :¼ 0.

L. Jin / Applied Mathematics and Computation 206 (2008) 186–192 191

Step 2: If EðzkÞ 6 e, then stop; otherwise, computer the search direction

dk ¼ �KðzkÞ�1/ðzkÞ:

Step 3: Computer the approximate iterate point

zkþ1 ¼ zk þ akdk;

where ak ¼ bik , ik is the smallest nonnegative integer i satisfying

Eðzk þ bidkÞ 6 ð1� 2qbiÞEðzkÞ:

Step 4: k :¼ kþ 1, go to Step 2.

Theorem 3.3. Let problem functions of problem (2.1) be twice continuously differentiable. Let fzkg be generated by Algorithm 3.1satisfying that for any cluster point z of this sequence, KðzÞ is nonsingular. Then the iterations fzkg converge to a solution toEðzÞ ¼ 0.

Proof. Obviously, we have that the sequence fEðzkÞg is monotone decreasing and convergent. Let �z be a cluster point of fzkg,then there is fzkj

g, such that

zkj! �z;

where fkjg# f1;2;3; . . .g. From Armijo line search rule, we have that

EðzkjÞ � Eðzkj

þ akj dkjÞP 2qakj Eðzkj

Þ;

EðzkjÞ � Eðzkj

þ akj�1dkjÞ < 2qakj�1Eðzkj

Þ:

Assume that Eð�zÞ–0, then from the first inequality above, we have for j!1, bkj ! 0. So from the second inequality, weobtain

Eðzkjþ akj�1dkj

Þ � EðzkjÞ

akj�1 > �2qEðzkjÞ;

hrEð�zÞ; �diP �2qEð�zÞ; ð3:4Þ

where �d ¼ �Kð�zÞ�1/ð�zÞ. Since

hrEð�zÞ; �di ¼ �h2Kð�zÞ/ð�zÞ;Kð�zÞ�1/ð�zÞi ¼ �2k/ð�zÞk2 ¼ �2Eð�zÞ; ð3:5Þ

by combining (3.4) and (3.5), we obtain that

0 6 ð2� 2qÞEð�zÞ 6 0;

therefore Eð�zÞ ¼ 0, which contradicts with the assumption Eð�zÞ–0. Therefore we must have Eð�zÞ ¼ 0. We conclude that anycluster point of fzkg is the solution of EðzÞ ¼ 0. h

Theorem 3.4. Let conditions of Theorem 2.1 be satisfied at the point ðx�;l�Þ, KðzÞ satisfies a Lipschitz condition in a neighborhoodof z�. Assume that the sequence fzkg generated by Algorithm 3.1 converges to z�, then there is an integer K large enough such thathk ¼ 1 for k > K, the sequence fzkg generated by Algorithm 3.1 with t > 0 converges quadratically to z�, wherel�i ¼ ðy�i Þ

2ði ¼ 1; . . . ;mÞ, z�T ¼ ðx�T; y�TÞ.

Proof. Let HðzÞ ¼ /ðzÞ, then Hðz�Þ ¼ 0nþm. From Lemma 3.1 we have that Kðz�Þ is a nonsingular matrix, then there are con-stants c0 > 0, c1 > 0, c2 > 0, c3 > 0, and a neighborhood Nðz�Þ, such that

kHðzÞk ¼ kHðzÞ � Hðz�ÞkP c0kz� z�k;kHðzÞk ¼ kHðzÞ � Hðz�Þk 6 c1kz� z�k;kKðzÞ �Kðz�Þk 6 c2kx� x�k;kKðzÞ�1k 6 c3; z 2Nðz�Þ

ð3:6Þ

for sufficiently large K, when k P K , we have

kzk � z�k 6 r0; zk 2Nðz�Þ; ð3:7Þ

where r0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2rp

c0=ðc1c3c2ð2þ c3c1ÞÞ, and therefore

kdkk ¼ kKðzkÞ�1HðzkÞk 6 c3c1kzk � z�k: ð3:8Þ

Table 1Numerical results

No. Algorithm IT k/ðzÞk2 Accuracy

1 [15, Tp100] R(3.1) 80 1:156948� 10�6 3:7042� 10�5

n ¼ 7, m ¼ 4 3.1 9 3:246312� 10�6 7:2006� 10�5

2 [15, Tp113] R(3.1) 296 7:028284� 10�7 2:2439� 10�5

n ¼ 10, m ¼ 8 3.1 77 3:861312� 10�6 7:1947� 10�4

3 [15, Tp45] R(3.1) 65 5:610742� 10�9 1:1439� 10�4

n ¼ 5, m ¼ 10 3.1 21 9:996834� 10�9 1:6342� 10�3

n stands for the dimension of variables, m for the number of constraints, IT for the number of iterations and R(3.1) for Runge–Kutta algorithm.

192 L. Jin / Applied Mathematics and Computation 206 (2008) 186–192

We can deduce from (3.6)–(3.8) that

kHðzk þ dkÞk ¼ kHðzk þ dkÞ � HðzkÞ þKðzkÞdkk 6 sup06t61

kðKðzk þ tdkÞ �KðzkÞÞdkk

6 sup06t61

kðKðzk þ tdkÞ �Kðz�ÞÞdkk þ kðKðzkÞ �Kðz�ÞÞdkk 6 2c2kzk � z�kkdkk þ c2kdkk2

6 c1c3c2ð2þ c3c1Þkzk � z�k2; ð3:9Þ

therefore, when k P K ,

EðzkÞ � Eðzk þ dkÞ � 2qEðzkÞ ¼ ð1� 2rÞkHðzkÞk2 � kHðzk þ dkÞk2

P ð1� 2qÞc20kzk � z�k2 � ½c1c3c2ð2þ c3c1Þ�2kzk � z�k4

P ½ð1� 2qÞc20 � ½c1c3c2ð2þ c3c1Þ�2r2

0�kzk � z�k2 P 0; ð3:10Þ

which implies that the step length hk ¼ 1 will satisfy Armijo line search rule when k P K in Algorithm 3.1 and hence Algo-rithm 3.1 becomes the classical Newton method and the rate of convergence is quadratic, see for instance [14]. h

4. Computation results

In this section, we gives three examples to illustrate the theoretical results achieved and the efficiency of Algorithm 3.1 inSection 3. The results are obtained by a preliminary MATLAB implementation of Algorithm 3.1 and the Runge–Kutta algo-rithm for the differential equation system (3.1). To compare their numerical results, we adopted the same initial pointand the parameter t in computing each of the problems. The numerical results given in Table 1 show that Runge–Kutta meth-od has better stability and higher precision than the Armijo stepsize method.

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