Differential systems for constrained optimization via a nonlinear augmented Lagrangian

Applied Mathematics and Computation 235 (2014) 482–491

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Differential systems for constrained optimization via a nonlinear augmented Lagrangian Li Jin a,⇑, Hongxia Yu b, Zhisong Liu a a b

Department of Mathematics, Zhejiang Ocean University, Zhejiang 316000, China School of Mathematics & Physics, University of Shanghai Electric Power, Shanghai 201300, China

a r t i c l e

i n f o

Keywords: Constrained optimization Nonlinear augmented Lagrangian Differential system Asymptotical stability Quadratic convergence

a b s t r a c t A general differential system framework for solving constrained optimization problems is investigated, which relies on a class of nonlinear augmented Lagrangians. The differential systems mainly consist of first-order derivatives and second-order derivatives of the nonlinear augmented Lagrangian. Under suitable conditions, the asymptotic stability of the differential systems and local convergence properties of their Euler discrete schemes are obtained, including the locally quadratic convergence rate of the discrete sequence for the second-order derivatives based differential system. Furthermore, as the special case, the exponential Lagrangian applied to this framework is given. Numerical experiments are presented illustrating the performance of the differential systems. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Consider the following optimization problems:

min f ðxÞ s:t: g i ðxÞ P 0;

i ¼ 1; . . . ; m:

ð1Þ

where f : Rn ! R and g i : Rn ! Rði ¼ 1; . . . ; mÞ are twice continuously differentiable functions. It is well known that the augmented Lagrange methods has many applications in the study of optimization problems. The first augmented Lagrangian, namely the proximal Lagrangian, was introduced by Rockafellar [1] and the theory of augmented Lagrangians were developed in, e.g., Ioffe [2], Bertsekas [3,4] and Rockafellar [5] for constrained optimization problems. Originally, the method was applied to problems with equality constraints [6,7] and later generalized to problems with inequality constraints [8,9]. By using augmented Lagrangian, constrained optimization problems can be translated into unconstrained optimization problems. The goal of this work is to apply differential equation methods for solving problem (1). The main idea of this type of methods is to construct the differential systems based on the nonlinear augmented Lagrangian. The corresponding systems mainly consist of first-order derivatives and second-order derivatives of the nonlinear augmented Lagrangian. It is demonstrated that the equilibrium point of this system coincides with the solution to the constrained optimization. The analysis of the methods is made on the basis of the stability theory of the solution of ordinary differential equations. In the original version of the differential equation methods was introduced by Arrow and Hurwicz [10], some results have been addressed in ⇑ Corresponding author. E-mail address: [email protected] (L. Jin). http://dx.doi.org/10.1016/j.amc.2014.02.097 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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L. Jin et al. / Applied Mathematics and Computation 235 (2014) 482–491

the work (see Refs. [11–13] for details). Among them, Evtushenko and Zhadan [14–18] have studied, by using the so-called space transformation techniques, a family of numerical methods for solving optimization problems with equality and inequality constraints. The proposed algorithms are based on the numerical integration of the systems of ordinary differential equation. Along this line, Zhang [19–21] studied modified versions of differential equation methods. This paper develops the results obtained in Ref. [22,23] motivated by finding a unified approach to the construction a family of differential systems to solve inequality constrained problems without using space transformations of Evtushenko and Zhadan. The method are based on a class of nonlinear augmented Lagranians, by which we construct first-order derivatives based and second-order derivatives based differential systems. The differential systems are carried out without using the space transformation and this feature provides a high rate of convergence. Under a set of suitable conditions, we prove the asymptotical stability of the two class of differential systems and local convergence properties of their Euler discrete schemes, including the locally quadratic convergence rate of the discrete sequence for second order derivatives based differential equation systems. Under this framework, two specific eases, the differential equation systems generated by the exponential Lagranian and the modified barrier function have been discussed in Ref. [22,23]. The paper is organized as follows. A class of nonlinear augmented Lagrangian and related properties are introduced in Section 2.1. Section 2.2 derives the differential system based on the nonlinear augmented Lagrangian and the asymptotic stability theorem is established under mild conditions. In Section 2.3, the Euler discrete schemes for the differential system is presented and the local convergence theorem is demonstrated. Under this framework, two specific cases, the differential systems generated by the exponential Lagrangian and the modified barrier function, are proposed in Section 2.4. In Section 3.1, we construct a second-order derivatives based differential system and prove the asymptotic stability of the system. In Section 3.2, Euler discrete schemes and their local convergence properties are obtained, including the locally quadratic convergence rate of the discrete sequence for the second order derivatives based differential system. The numerical results show that Runge–Kutta method has better stability and higher precision in Section 4. 2. Differential systems based on first-order derivatives In this section, we mention some preliminaries that will be used throughout this paper. Without loss of generality, we assume that f ðxÞ and g i ðxÞ; i ¼ 1; . . . ; m are twice continuously differentiable, then the classical Lagrangian for problem P (1) defined by Lðx; kÞ ¼ f ðxÞ  m i¼1 ki g i ðxÞ, for any feasible point x, the active set of indices is denoted by IðxÞ ¼ fijg i ðxÞ ¼ 0; i ¼ 1; . . . ; mg. Let x be a local minimizer to problem (1) and the pair ðx ; k Þ be the corresponding KKT point, which satisfies the following conditions: m X

rx Lðx ; k Þ ¼ rf ðx Þ 

ki rg i ðx Þ ¼ 0;

ki P 0;

ki g i ðx Þ ¼ 0;

g i ðx Þ P 0;

i ¼ 1; . . . ; m:

ð2Þ

i¼1

Let the Jacobian uniqueness conditions, proposed in [11], hold at ðx ; k Þ: (1) The multipliers k > 0; i 2 Iðx Þ. (2) The gradients rg i ðx Þ; i 2 Iðx Þ are linearly independent. (3) yT r2xx Lðx ; k Þy > 0; 80 – y 2 fyjrg i ðx ÞT y ¼ 0; i 2 Iðx Þg. 2.1. The properties of nonlinear augmented Lagrangians We introduce a class of nonlinear augmented Lagrangian for problem (1), in the following form

Fðx; y; tÞ ¼ f ðxÞ  jtj

m X

Wðjtj1 g i ðxÞ; yi Þ;

ðt – 0Þ;

i¼1

where t is a controlling parameter, Wðw; v Þ : R2 ! R1 is a real-value function and yi ði ¼ 1; . . . ; mÞ are variables of the nonlinear Lagrangian multiplier vector. By using nonlinear augmented Lagrangian Fðx; y; tÞ, the differential systems for solving problem (1) will be derived in the next section. The following lemma will be used in the proof of the forthcoming theorem. m

Lemma 2.1 (See Ref. [3]). Let A be a n  n symmetrical matrix, B be a r  n matrix, U ¼ ½diag li i¼1 , where l ¼ ðl1 ; . . . ; lr Þ > 0. If k > 0 is a scalar and By ¼ 0 implies hAy; yi P khy; yi. Then there are scalars k0 > 0 and c 2 ð0; kÞ such that, for any k P k0 , T

hðA þ kB UBÞx; xi P chx; xi;

8x 2 R n :

Now we discuss the properties of nonlinear augmented LagrangianFðx; y; tÞ. Theorem 2.1. Let ðx ; k Þ be a KKT point of (1), the Jacobian uniqueness conditions hold at ðx ; k Þ, and the following conclusions are satisfied:

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L. Jin et al. / Applied Mathematics and Computation 235 (2014) 482–491

ð1Þ

@W ð0; v Þ ¼ v ; @w

ð2Þ

@2W ðw; v Þ < 0; @w2

@W ðw; 0Þ ¼ 0; @w

v > 0;

w > 0:

@2W ðw; 0Þ ¼ 0; @w2

v > 0;

w > 0:

then there exists y ¼ k such that (I) rx Fðx ; y ; tÞ ¼ 0n ; t – 0 (II) 9^t > 0 and c > 0, for any jtj 2 ð0; ^tÞ, such that

hr2xx Fðx ; y ; tÞw; wi P chw; wi;

8w 2 Rn :

Proof. Without loss of generality, we assume that Iðx Þ ¼ f1; . . . ; rg, where r 6 m. We compute the gradient rx Fðx ; y ; tÞ of Fðx; y; tÞ at ðx ; y ; tÞ

rx Fðx ; y ; tÞ ¼ rf ðx Þ 

m X @W i¼1

@w

ðjtj1 g i ðx Þ; yi Þrg i ðx Þ:

ð3Þ

By part 1 of Theorem 2.1, @ W=@wðjtj1 g i ðx Þ; yi Þ ¼ yi ; ki ¼ yi 1 6 i 6 m, we have that

rx Fðx ; y ; tÞ ¼ rf ðx Þ 

X

ki rg i ðx Þ 

i2Iðx Þ

X

0  rg i ðx Þ ¼ rf ðx Þ 

iRIðx Þ

m X ki rg i ðx Þ ¼ rx Lðx ; k Þ ¼ 0: i¼1

Hence the first condition in Theorem 2.1 is true. Similarly, by calculating, we have that

r2xx Fðx ; y ; tÞ ¼ r2 f ðx Þ 

m X @W i¼1

@w

ðjtj1 g i ðx Þ; yi Þr2 g i ðx Þ  jtj1

m X @2W i¼1

@w2

ðjtj1 g i ðx Þ; yi Þrg i ðx Þrg i ðx ÞT :

Since the part 2 of Theorem 2.1 holds, it follows that m X @ 2 W 1 X ðjtj g i ðx Þ; yi Þrg i ðx Þrg i ðx ÞT  jtj1 0  rg i ðx Þrg i ðx ÞT 2 @w   i2Iðx Þ iRIðx Þ ! 2 X @ W 1 ¼ r2xx Lðx ; k Þ þ jtj1  ðjtj g i ðx Þ; yi Þ rg i ðx Þrg i ðx ÞT : @w2 i2Iðx Þ

r2xx Fðx ; y ; tÞ ¼ r2xx Lðx ; k Þ  jtj1

In view of the part 2 of Theorem 2.1, we obtain

jtj1

X i2Iðx Þ



@ 2 W 1 ðjtj g i ðx Þ; yi Þ @w2

! >0

Since Lemma 2.1 and the Jacobian uniqueness conditions, we must have (II), The proof is completed. h 2.2. The asymptotic stability of the differential system To obtain the numerical solution of problem (1), we seek the limit point of the solutions of the system described by the following vector differential equation

3 m X 1 @W  r f ðxÞ þ ðjtj g ðxÞ; y Þ r g ðxÞ 7 6 i i i @w 7 6 i¼1 7 6 7 6 1 @ W 7 6 jtj ðjtj g ðxÞ; y Þ 1 1 @v dz 6 7 ¼6 7; 1 @W 7 6 jtj ðjtj g ðxÞ; y Þ dt 2 2 @ v 7 6 7 6 .. 7 6 . 5 4 1 jtj @@Wv ðjtj g m ðxÞ; ym Þ 2

ðt – 0Þ:

where zT ¼ ðxT ; yT Þ. We will investigate the local behavior of trajectories of system (4) in the neighborhood of the point x . Theorem 2.2. Let conditions of Theorem 2.1 and the following conditions be satisfied at the point ðx ; k Þ,

ð4Þ

L. Jin et al. / Applied Mathematics and Computation 235 (2014) 482–491

ð1Þ

@2W ð0; v Þ – 0; @w@ v

ð2Þ

@2W ð0; v Þ ¼ 0; @v 2

@2W ðw; 0Þ ¼ 0; @w@ v

v > 0; v > 0;

@2W ðw; 0Þ > 0; @v 2

485

w > 0:

w > 0:

Then, 9^t > 0 and c > 0, for any jtj 2 ð0; ^tÞ, the system (4) is asymptotically stable at ðx ; y Þ, where y ¼ k . Proof. Without loss of generality, we assume that Iðx Þ ¼ f1; . . . ; rg with r 6 m. Linearizing system (2.3) in the neighborhood of zT ¼ ðxT ; yT Þ, where y ¼ k , we obtain

dz ¼ Q ðz  z Þ; dt

ð5Þ

and where

Q¼



Q 11

Q 12

Q 21

Q 22

 ;

Q 11 ¼ r2xx Lðx ; k Þ  jtj1 0 Q 22

B B ¼B @

Q 12 ¼ Q 21 ¼

r X @2W i¼1

@w2

ðjtj1 g i ðx Þ; li Þrg i ðx Þrg i ðx ÞT ; 1

2

jtj @@ vW2 ðjtj1 g 1 ðx Þ; l1 Þ ..

. 2

jtj @@ vW2 ðjtj1 g m ðx Þ; lm Þ

C C C; A

! @2W @2W ðjtj1 g 1 ðx Þ; l1 Þrg 1 ðx Þ; . . . ;  ðjtj1 g m ðx Þ; lm Þrg m ðx Þ ; @w@ v @w@ v ! 2 2 @ W @ W ðjtj1 g 1 ðx Þ; l1 Þrg 1 ðx ÞT ; . . . ; ðjtj1 g m ðx Þ; lm Þrg m ðx ÞT : @w@ v @w@ v 

From Theorem 2.1 it follows that

Q 11 ¼ r2xx Fðx ; l ; tÞ;

Q 12 ¼ Q T21 :

The stability of system (4) is determined by the properties of the roots of the characteristic equation

detðQ  kInþm Þ ¼ 0:

ð6Þ

^ denote the conjugate vector of a complex vector y, ReðbÞ denote the real part of the complex number b. Let a be an Let y eigenvalue of Q and ðzT1 ; zT2 Þ – 0, where z1 2 Rn and z2 2 Rm , be the corresponding nonzero eigenvector of the matrix Q. Then

      z1 z1 ¼ Re a½z^1 T ; z^2 T  ¼ ReðaÞðjz1 j2 þ jz2 j2 Þ: Re ½z^1 T ; z^2 T Q z2 z2

ð7Þ

In view of the definition of Q, we have

   z1 ¼ Refz^1 T r2xx Fðx ; l ; tÞz1 þ z^2 T Q 21 z1 þ z^1 T Q 12 z2 þ z^2 T Q 22 z2 g: Re ½z^1 T ; z^2 T Q z2

ð8Þ

Since for any real matrix M, it follows that

^ T MT zg Ref^zT M T wg ¼ Refw Combining Q 21 ¼

Q T12 ,

ð9Þ

(9) and (8), we obtain

   z1 ¼ Refz^1 T r2xx Fðx ; l ; tÞz1 þ z^2 T Q 22 z2 g: Re ½z^1 T ; z^2 T Q 1 z2

ð10Þ

The above relation and (7) imply that

ReðaÞðjz1 j2 þ jz2 j2 Þ ¼ Refz^1 T r2xx Fðx ; l ; tÞz1 þ z^2 T Q 22 z2 g: 2  xx Fðx ;

Since the matrix r

ð11Þ

l ; tÞ is positive definite, we obtain

Refz^1 T r2xx Fðx ; l ; tÞz1 þ z^2 T Q 22 z2 g > 0;

z1 – 0

ð12Þ

We also have by using (10) and (12),

ReðaÞ > 0 or z1 ¼ 0:

ð13Þ

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L. Jin et al. / Applied Mathematics and Computation 235 (2014) 482–491

If z1 ¼ 0,

 Q

z1



z2

¼a



z1 z2

 :

ð14Þ

We have

 ð1!rÞ

Let z2

! @2W @2W ðjtj1 g 1 ðx Þ; l1 Þrg 1 ðx Þ; . . . ;  ðjtj1 g m ðx Þ; lm Þrg m ðx Þ z2 ¼ 0: @w@ v @w@ v T

¼ ðz12 ; z22 ; . . . ; zr2 Þ , then the above relation can also be written as



! @2W @2W ð1rÞ ðjtj1 g 1 ðx Þ; l1 Þrg 1 ðx Þ; . . . ;  ðjtj1 g r ðx Þ; lr Þrg r ðx Þ z2 ¼ 0: @w@ v @w@ v

From the Jacobian uniqueness conditions, we have ð1!rÞ

z2

¼ 0:

ð15Þ

By using the part 2 of Theorem 2.2 and (14), we have

0 B B B B B B B B @

10

0 ..

. 2

jtj @@ vW2 ðjtj1 g rþ1 ðx Þ; lrþ1 Þ ..

. 2

jtj @@ vW2 ðjtj1 g m ðx Þ; lm Þ

0 1 0 0 1 CB . C B . C CB .. C B .. C CB C B CB rþ1 C B rþ1 C CB z2 C C ¼ B z2 C; CB C B CB . C B . C C@ .. A @ .. C A A zm 2

zm 2

which is also expressed as

jtj

@ 2 W 1 ðjtj g j ðx Þ; lj Þzj2 ¼ zj2 ; @v 2

j ¼ r þ 1; . . . ; m:

and implies that

jtj

@ 2 W 1 ðjtj g j ðx Þ; lj Þ ¼ 1; @v 2

or zj2 ¼ 0;

j ¼ r þ 1; . . . ; m:

ð16Þ

On the other hand, since jtj 2 ð0; ^tÞ, we have ðrþ1!mÞ

z2

¼ 0;

ð17Þ zT2

and by using (15) and (17), we obtain ¼ 0, and from the assumption we have T ðzT1 ; zT2 Þ – 0. So we must have z1 – 0. From (13) we have

T ðzT1 ; zT2 Þ

¼ 0. This contradicts with

ReðaÞ > 0: Therefore, we have that all eigenvalues of Q have negative real parts. It follows from Lyapunov’s stability theorem of the first-order approximation that ðx ; y Þ is a local asymptotically stable equilibrium point of (4). h 2.3. Euler discrete schemes Integrating the system (4) by the Euler method, one obtains the iterate process

xkþ1 ¼ xk  hk rf ðxk Þ 

m X @W i¼1

yðkþ1Þi ¼ yki  hk jtj

@w

! ðjtj1 g i ðxk Þ; yki Þrg i ðxk Þ

ð18Þ

@ W 1 ðjtj g i ðxk Þ; yki Þ i ¼ 1; 2; . . . ; m: @v T

where zTk ¼ ðxTk ; yTk Þ; yk ¼ ðy1k ; y2k ; . . . ; ym k Þ and hk is a stepsize. The following theorem tells us that the Euler discrete schemes with a constant stepsize is locally convergent.  > 0 such that for any Theorem 2.3. Let conditions of Theorem 2.2 be satisfied at the point ðx ; k Þ. Then there exists a h  the iterations defined by (18) with t > 0 converge locally to ðx ; y Þ, where y ¼ k ; i ¼ 1; . . . ; m. 0 < hk < h, i i Proof. Let zT ¼ ðxT ; yT Þ, since z is a fixed point of the mapping

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L. Jin et al. / Applied Mathematics and Computation 235 (2014) 482–491

0 B rf ðxÞ 

B B B B B RðzÞ ¼ z  hB B B B B @

m X

@W ðjtj1 g i ðxÞ; @w

i¼1

jtj @@Wv ðjtj1 g 1 ðxÞ; jtj @@Wv ðjtj1 g 2 ðxÞ;

1

li Þrg i ðxÞ C

l1 Þ l2 Þ

.. . jtj @@Wv ðjtj1 g m ðxÞ; lm Þ

C C C C C C; C C C C A

ð19Þ

 can be chosen such that the iterations deThe convergence of the iterations (18) will be proved if we demonstrate that a h fined by

zkþ1 ¼ Fðzk Þ  Let rFðzÞ be Jacobian transpose of FðzÞ and m1 ; . . . ; mnþm converge to z whenever z0 is in a neighborhood of z and 0 < hk < h. be the eigenvalues of the matrix rFðz ÞT with the expression

mj ¼ ð1  haj Þ  iðhbj Þ: From the proof of Theorem 2.2, we have aj > 0. The condition jmj j < 1 can be written as 2

h < 2aj =ða2j þ bj Þ;

1 6 j 6 n þ m:

Let

 ¼ minf2a =ða2 þ b2 Þj1 6 j 6 n þ mg; h j j j  and the iterations generated by the scheme (2.12) is locally then the spectral radius of rFðz ÞT is strictly less than 1 for h < h,   convergent to ðx ; y Þ (see Evtushenko [15]). The proof is completed. h 2.4. Examples 2 r g i ðxÞ To illustrate this systems, we respectively used the exponential Lagrangian F r ðx; yÞ ¼ f ðxÞ  rRm  1Þ and the i¼1 yi ðe P 1 2 y lgðt g ðxÞ þ 1Þ, then the system (4) has the following forms modified barrier function F t ðx; yÞ ¼ f ðxÞ  t m i i¼1 i 1

3 m X 2 r1 g i ðxÞ r f ðxÞ þ y e r g ðxÞ  i i 7 6 7 6 i¼1 7 6 1 7 6 r g1 ðxÞ  1Þ r y ðe 2 7 dz 6 1 7 ¼6 1 7 dt 6 2ry2 ðer g2 ðxÞ  1Þ 7 6 7 6 .. 7 6 5 4 . 2

2rym ðer

1 g

m ðxÞ

ð20Þ

 1Þ

and

2

m X y2i rg i ðxÞ 6 rf ðxÞ þ t 1 g i ðxÞþ1 6 i¼1 dz 6 6 2ty lgðt1 g ðxÞ þ 1Þ ¼6 1 1 dt 6 .. 6 4 .

3 7 7 7 7 7; 7 7 5

ð21Þ

2tym lgðt1 g m ðxÞ þ 1Þ where r > 0; t > 0. In our first publication, the asymptotic stability of systems (20,21) and local convergence properties of their Euler discrete schemes are obtained, and the algorithms with Armijo line searches and the Runge–Kutta algorithms for the two systems (20,21) are employed to solve several numerical examples (see Ref. [22,23] for more details). 3. Differential systems based on second-order derivatives The method in Section 2 can be viewed as a gradient approach, through the system (4), for solving problem (1). Now we discuss the Newton version. The continuous version of Newton method leads to the initial value problem for the following system of ordinary differential equations

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0 B rf ðxÞ 

B B B B dz B KðzÞ ¼  diag ci B dt 16i6nþm B B B B @

m X

@W ðjtj1 g i ðxÞ; yi Þ @w

i¼1

1

rg i ðxÞ C

C C C C C C; C C C C A

jtj @@Wv ðjtj1 g 1 ðxÞ; y1 Þ jtj @@Wv ðjtj1 g 2 ðxÞ; y2 Þ .. . jtj @@Wv ðjtj1 g m ðxÞ; ym Þ

ð22Þ

where zT ¼ ðxT ; yT Þ; ðc1 ; . . . ; cnþm ÞT > 0 is a scaling vector, KðzÞ is the Jacobian matrix of the following mapping

1 m X 1 @W ðjtj g ðxÞ; y Þ r g ðxÞ C B rf ðxÞ  i i i @w C B i¼1 C B C B C B jtj @@Wv ðjtj1 g 1 ðxÞ; y1 Þ C B /ðzÞ ¼ B C; 1 @W C B jtj @ v ðjtj g 2 ðxÞ; y2 Þ C B C B . C B .. A @ 1 @W jtj @v ðjtj g m ðxÞ; ym Þ 0

ð23Þ

and can be express as



KðzÞ ¼

 KðzÞ11 KðzÞ12 ; KðzÞ21 KðzÞ22

where KðzÞ11 2 Rnn ; KðzÞ12 2 Rnm ; KðzÞ21 2 Rmn ; KðzÞ22 2 Rmm . We can easily get the formula,

KðzÞ11 ¼ r2 f ðxÞ  

KðzÞ12 KðzÞ21

KðzÞ22

m X @W i¼1

@w

ðjtj1 g i ðxÞ; yi Þr2 g i ðxÞ  jtj1

m X @2W i¼1

@w2

ðjtj1 g i ðxÞ; yi Þrg i ðxÞrg i ðxÞT ;

 @W @W @W ¼  ðjtj1 g 1 ðxÞ; y1 Þrg 1 ðxÞ;  ðjtj1 g 2 ðxÞ; y2 Þrg 2 ðxÞ; . . . ;  ðjtj1 g m ðxÞ; ym Þrg m ðxÞ ; @w@ v @w@ v @w@ v   @W @ W @ W ¼ ðjtj1 g 1 ðxÞ; y1 Þrg 1 ðxÞT ; ðjtj1 g 2 ðxÞ; y2 Þrg 2 ðxÞT ; . . . ; ðjtj1 g m ðxÞ; ym Þrg m ðxÞT ; @w@ v @w@ v @w@ v 0 1 1 @2 W jtj 2 ðjtj g 1 ðxÞ; y1 Þ B @v C B C .. ¼B C: . @ A 2 jtj @@vW2 ðjtj1 g m ðxÞ; ym Þ

Lemma 3.1. Let conditions of Theorem 2.2 be satisfied at the point ðx ; k Þ. Then, 9^t > 0, for jtj 2 ð0; ^tÞ; Kðz Þ is a nonsingular matrix, where yi ¼ ki ði ¼ 1; . . . ; mÞ and zT ¼ ðxT ; yT Þ. Proof. Let zT ¼ ðxT ; yT Þ, the matrix Kðz Þ coincides with the matrix Q introduced in (5). It was shown in the proof of Theorem 2.2 that all eigenvalues of this matrix are strictly positive. Therefore, Kðz Þ is nonsingular. h 3.1. The asymptotic stability of system (22)

Theorem 3.1. Let conditions of Theorem 2.2 be satisfied at the point ðx ; k Þ. Then, 9^t > 0, for jtj 2 ð0; ^tÞ, system (22) is asymptotically stable at z , where zT ¼ ðxT ; yT Þ and yi ¼ ki ði ¼ 1; . . . ; mÞ. Proof. From the second order smoothness of /ðzÞ around z , we have

/ðzÞ ¼ /ðz Þ þ Kðz ÞdðzÞ þ HðdðzÞÞ; where dðzÞ ¼ z  z ; HðdðzÞÞ ¼ OðkdðzÞk2 Þ. Linearizing system (22) at the point z , we obtain

ddðzÞ ^ ðz ÞdðzÞ; ¼ Q dt

 ^ ðz Þ ¼ Kðz Þ1 diag Q 16i6nþm ci Kðz Þ:

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^ ðz Þ is similar to matrix diag Matrix Q 16i6nþm ci ; therefore, they have the same eigenvalues ki ¼ ci > 0; 1 6 i 6 n þ m. According to Lyapunov linearization principle, we have that the equilibrium point z is asymptotically stable. h 3.2. Euler discrete schemes for system (22) Integrating system (22) by the Euler method, one obtains the iterate process

xkþ1 ¼ xk  hk rf ðxk Þ 

m X @W i¼1

lðkþ1Þi ¼ lki  hk jtj

@w

! ðjtj1 g i ðxk Þ; lki Þrg i ðxk Þ

ð24Þ

@ W 1 ðjtj g i ðxk Þ; lki Þi ¼ 1; 2; . . . ; m: @v T

where zTk ¼ ðxTk ; yTk Þ; yk ¼ ðy1k ; y2k ; . . . ; ym k Þ and hk is a stepsize. Theorem 3.2. Let conditions of Theorem 2.2 be satisfied at the point ðx ; k Þ. Then, 9^t > 0, for jtj 2 ð0; ^tÞ, the discrete version fzk g defined by (24) locally converges with at least linear rate to the point z if the stepsize hk is fixed and hk < 2=max16i6nþm ci . If hk ¼ 1 and ci ¼ 1; 1 6 i 6 n þ m, then the sequence fzk g converges quadratically to z , where zT ¼ ðxT ; yT Þ and yi ¼ ki ði ¼ 1; . . . ; mÞ. Proof. The proof of this Theorem is nearly identical to the proof of Theorem 2.3 and to the proof of convergence of Newton’s method. h

4. Numerical illustration we respectively used the exponential Lagrangian and the modified barrier function, then the system (22) has the following forms

3 m X 1 y2i er gi ðxÞ rg i ðxÞ 7 6 rf ðxÞ  7 6 i¼1 7 6 7 6 1 g ðxÞ r 1 7 6 2ry1 ðe  1Þ dz 7 6 KðzÞ ¼  diag ci 6 7; 1 g ðxÞ r 2 7 dt  1Þ 2ry2 ðe 16i6nþm 6 7 6 7 6 . 7 6 . . 5 4 2

2rym ðer

1 g

m ðxÞ

ð25Þ

 1Þ

and

2 6 rf ðxÞ 

m X

y2i rg i ðxÞ t 1 g i ðxÞþ1

6 i¼1 6 6 dz 1 2ty1 lgðt g 1 ðxÞ þ 1Þ KðzÞ ¼ diag16i6nþm ci 6 6 dt 6 .. 6 . 4

3 7 7 7 7 7: 7 7 7 5

ð26Þ

2tym lgðt 1 g m ðxÞ þ 1Þ The convergence rate results and analysis of systems (25, 26) have been discussed in Ref. [22,23]. Based on system (25), we report numerical experiments for the testing problems (Tp100 and Tp113 are from Ref. [24]) to illustrate the theoretical results achieved and the efficiency in Section 3. The results in Table 1 are obtained by a preliminary MATLAB 7.1 implementation of the Runge–Kutta algorithm for the differential system (25), where n denotes the dimension of variables, m denotes the number of constraints, IT denotes the number of iterations, and SðzÞ ¼ k/ðzÞk2 is in (23). We work out the details of these results in figures, Fig. 1 shows the local behavior of trajectories of system (25), Fig. 2 exhibits a substantially reduced

Table 1 Numerical results. Test

n

m

IT

Tp113 [24]

10

8

72

SðzÞ

accuracy

1.556631 10

Tp100 [24]

7

4

180

1.0373 104

6

8.2329 10 0.6889

time(s) 5

85.499 204.411

490

L. Jin et al. / Applied Mathematics and Computation 235 (2014) 482–491 10

5

9 4

8 7

3 z(t)

z(t)

6 5

2

4 1

3 2

0

1 0

0

2

4

t

6

8

−1

10

0

2

4

6 t

8

10

12

Fig. 1. The local behavior of trajectories of problem Tp113 (left) and Tp100 (right) based on system (25).

6

4

x 10

3

x 10

10

2.5

8

2 S(z)

S(z)

12

6

1.5

4

1

2

0.5

0 0

10

20

30

40 IT

50

60

70

80

0

0

50

100 IT

150

200

Fig. 2. Cost function and the number of iterations: Tp113 (left) and Tp100 (right).

variability in terms of the objective value as the number of iterations increased. The numerical results given show that the differential equation methods has better stability and higher precision. 5. Conclusion In this study, using a class of nonlinear augmented Lagrangians, we construct a first order derivatives based and a second order derivatives based differential equation systems for inequality constrained optimization problems. Under a set of suitable conditions, we prove the asymptotic stability of the two differential systems and local convergence properties of their Euler discrete schemes, including the locally quadratic convergence rate of the discrete algorithm for second order derivatives based differential equation system. Under this framework, two specific eases, the differential equation systems generated by the exponential Lagrangian and the modified barrier function, are discussed. In a subsequent paper we will consider extension of the method in application to problems of interest in science and engineering. Acknowledgment The work was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ12A01007. References [1] R.T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming, SIAM J. Control 12 (2) (1974) 68–285. [2] A. Ioffe, Necessary and sufficient conditions for a local minimum 3: second-order conditions and augmented duality, SIAM J. Control Optim. 17 (1979) 266–288. [3] D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982. [4] D.P. Bertsekas, Nonlinear Programming, Athena Scientific, 1999. [5] R.T. Rockafellar, Lagrange multipliers and optimality, SIAM Rev. 35 (1993) 183–238. [6] M. Hestenes, Multiplier and gradient methods, J. Optim. Theor. Appl. 4 (1969) 303–320.

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