Applied Mathematics and Computation 215 (2009) 2521–2527

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Applied Mathematics and Computation

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

Inexact non-interior continuation method for solving large-scalemonotone SDCP q

Shao-Ping Rui a,b,*, Cheng-Xian Xu a

a Department of Mathematics, Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, PR Chinab Department of Mathematics, Huaibei Coal Industry Teachers College, Huaibei 235000, PR China

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

Keywords:Monotone semidefinite complementarityproblemInexact non-interior continuation methodLarge-scale problemLocal superlinear convergence

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.08.059

q This work is supported by National Natural Scie* Corresponding author. Address: Department of

E-mail addresses: rsp9999@163.com (S.-P. Rui), m

For exact Newton method for solving monotone semidefinite complementarity problems(SDCP), one needs to exactly solve a linear system of equations at each iteration. For prob-lems of large size, solving the linear system of equations exactly can be very expensive. Inthis paper, we propose a new inexact smoothing/continuation algorithm for solution oflarge-scale monotone SDCP. At each iteration the corresponding linear system of equationsis solved only approximately. Under mild assumptions, the algorithm is shown to be bothglobally and superlinearly convergent.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

In the last serval years, the semidefinite programming problem (SDP) have been studied intensively [3,15–18]. This prob-lem has important applications [9,10], which often call for the solution of large-scale problems. As a natural generalization,the semidefinite complementarity problem (SDCP) is defined as follows [1]: for a given mapping F : S!S#X, find anðx; yÞ 2S�S satisfying

x 2Sþ; y 2Sþ; hx; yi ¼ 0; FðxÞ ¼ y; ð1:1Þ

where X denote the space of n� n block-diagonal real matrices with m block of sizes n1; . . . ;nm, respectively (the block arefixed), S denote the subspace comprising those x 2 X that are symmetric, that is, xT ¼ x; Sþ denote the convex cone com-prising those x 2S that are positive semidefinite, F is a mapping from S into itself, and h�; �i is the inner product defined byhx; yi :¼ tr½xT y�, where tr½�� denotes the matrix trace. Similar to [1], in this paper, we assume that F is continuously differen-tiable monotone function.

A few methods have been developed to solve this problem, such as interior point methods [12], merit function methods[11], and non-interior continuation methods [1,13]. We are interested in non-interior continuation methods for solvingmonotone SDCP. The pioneer work was done by Chen and Tseng [1]: they studied the existence of Newton directions andboundedness of iterates, and then extended the non-interior continuation method for the NCP to monotone SDCP. The globallinear and local superlinear convergence results of the algorithm were proved and the promising numerical results were re-ported in their paper. After Chen and Tsengs encouraging work, several algorithms and theoretical results for SDCP have beendeveloped [5,13,14].

In exact Newton non-interior continuation method, each iteration consists of finding a solution of linear system of equa-tions which may be expensive if one is solving a large-scale problem. A lot of inexact Newton methods have been proposed

. All rights reserved.

nce Foundations of China 10671152.Mathematics, Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, PR China.

xxu@mail.xjtu.edu.cn (C.-X. Xu).

2522 S.-P. Rui, C.-X. Xu / Applied Mathematics and Computation 215 (2009) 2521–2527

for solving nonlinear complementarity problem (NCP) [2,7]. In this paper, we will extend inexact Newton methods for solv-ing NCP to large-scale SDCP under the framework of non-interior continuation method. In such an inexact method, the linearsystem of equations is solved only up to a certain degree of accuracy. The accuracy level of approximate solution is controlledby the so-called forcing parameter which links the norm of residual vector to the norm of mapping at the current iterate. Weshow that, under mild assumptions, our algorithm is locally superlinearly and even quadratically convergent to a solution ofthe monotone SDCP.

The paper is organized as follows: in Section 2 we present an inexact non-interior continuation algorithm for solvinglarge-scale monotone SDCP. Convergence results are analyzed in Section 3. Conclusions are given in Section 4.

Throughout this paper, we use the following notation. ‘‘:¼” means ‘‘is defined as”. We denote byrFðxÞ the Jacobian of F ateach x 2S, viewed as a linear mapping from S to S. Sþþ denote the convex cone comprising those x 2S that are positivedefinite. Rþ and Rþþ denote the nonnegative and positive reals. We write x � y to mean x� y is positive definite. Landau sym-bols oð�Þ and Oð�Þ are defined in usual way. For matrices x 2S; kxk is the Frobenius norm. For vector a 2 Rn; kak denotes 2-norm.

2. Algorithm description

Our inexact non-interior continuation method is based on the following smoothed Fischer–Burmeister function:

/lðx; yÞ ¼ xþ y� ðx2 þ y2 þ 2l2IÞ12; ð2:1Þ

where ðl; x; yÞ 2 R�S�S and I is the n� n identity matrix. This smoothing function was introduced by Kanzow [19] in thecase of the NCP based on the Fischer–Burmeister function [20]. Let

Hlðx; yÞ :¼/lðx; yÞFðxÞ � y

� �: ð2:2Þ

From Lemma 1 of [1], we know that if l! 0, then

Hlðx; yÞ ! H0ðx�y�Þ :¼ x� � ½x� � y��þ;

where ½x� � y��þ denotes the orthogonal projection of x� � y� at Sþ. Then, by Lemma 2.1 of [11],

H0ðx�y�Þ ¼ 0() ðx�y�Þ solves monotone SDCP:

In the remainder of this paper, we will view the number l as an independent variable. In order to make this clear in ournotation, we set /ðl; x; yÞ ¼ /lðx; yÞ, that is

/ðl; x; yÞ ¼ xþ y� ðx2 þ y2 þ 2l2IÞ12: ð2:3Þ

Further, let

Hðl; x; yÞ :¼l

FðxÞ � y

/ðl; x; yÞ

0B@1CA: ð2:4Þ

In addition, we endow the vector space R�S�S with the norm

jjjðl; x; yÞjjj :¼ ðjlj2 þ kxk2 þ kyk2Þ12:

Define the merit function h : Rþ �S�S! Rþ by

hðl; x; yÞ ¼ jjjHðl; x; yÞjjj2 ¼ ðlÞ2 þ kFðxÞ � yk2 þ k/ðl; x; yÞk2:

In the remainder of this paper, for the sake of simplicity, denote z :¼ ðl; x; yÞ; zk :¼ ðlk; xk; ykÞ; Dzk ¼: ðDlk;Dxk;DykÞ. In eachiteration of exact non-interior continuation method for monotone SDCP, linear system of equations

rHðzkÞðDzkÞ ¼ �HðzkÞ ð2:5Þ

must be solved exactly, whererHðzkÞ denote the Jacobian of H at zk. However, in each iteration of our method (2.5) is solvedonly approximately. Our method is to solve the following system of equations

rHðzkÞðDzkÞ ¼ �HðzkÞ þ lbðzkÞrk

!; ð2:6Þ

where bðzkÞ ¼ cminf1;hðzkÞg; c 2 ð0;1Þ is a constant, l 2 Rþþ is a constant satisfying cl < 1, krkk 6 gkjjjHðlk; xk; ykÞjjj;gk 2 ð0;1Þ. As it is usual in the context of inexact methods, we will refer to the vector rk as the residual vector and tothe parameter gk as the forcing term. The forcing term is used to control the level of accuracy in solving the systems equa-tions. We would like to remark that the criterion krkk 6 gkjjjHðlk; xk; ykÞjjj allows us to solve (2.5) to a very low accuracy

S.-P. Rui, C.-X. Xu / Applied Mathematics and Computation 215 (2009) 2521–2527 2523

when the iterates are far from the solution, that is, when kHðlk; xk; ykÞk is large. The neighborhood used in our paper isgiven by

NðlÞ ¼ fzk 2 Rþþ �S�S : lbðzkÞ 6 lkg: ð2:7Þ

Let v be a function from ð0;1� clÞ to ð0;1� clÞ. We are now to give a formal statement of our inexact non-interior contin-uation method for the solution of large-scale monotone SDCP.

Algorithm 2.1. Choose constants d 2 ð0;1Þ; c 2 ð0;1Þ and r 2 ð0;1=2Þ. Let ðx0; y0Þ 2S�S be an arbitrary point. Takel0 ¼ l, z0 ¼ ðl0; x0; y0Þ; g0 2 ð0;1� clÞ and k :¼ 0.

Step 1. If hðzkÞ ¼ 0, stop.Step 2. Compute Dzk by (2.6).Step 3. Let hk be the maximum of the values 1; d; d2; . . . such that

hðzk þ hkDzkÞ 6 pðhkÞhðzkÞ; ð2:8Þ

where pðhkÞ ¼ ½1� rð1� cl� gkÞhk� 2 ð0;1Þ.Step 4. Set zkþ1 ¼ zk þ hkDzk;gkþ1 ¼ vðgkÞ and k ¼ kþ 1, go to Step 1.

Remark 2.1. (i) Algorithm 2.1 can be started easily. In fact, from l0 ¼ l and the fact bðzkÞ < 1, we have z0 2NðlÞ. This indi-cates initial point z0 is arbitrary. (ii) In theory we use hðzkÞ ¼ 0 as a termination of our algorithm. In practice, however, we usehðzkÞ 6 e as a termination rule, where e is a pre-set tolerance error.

To verify that Algorithm 2.1 is well-defined, we need the following three lemmas.

Lemma 2.1 [3]. Let ðl; x; yÞ 2 Rþ �S�S and /ðl; x; yÞ is given by (2.3). If x2 þ y2 þ 2l2I 2Sþþ, then /ðl; x; yÞ iscontinuously differentiable in ðl; x; yÞ. In particular, /ðl; x; yÞ is continuously differentiable at any ðl; x; yÞ 2 Rþþ �S�S.

Lemma 2.2. [1] If F is a continuously differentiable monotone function, then rHðzÞ is nonsingular for all z 2 Rþþ �S�S.

Lemma 2.3. Suppose that F is a continuously differentiable monotone function. For any �z :¼ ð�l; �x; �yÞ 2 Rþþ �S�S, then thereexists a closed neighborhood Bð�zÞ of �z and a positive number �h 2 ð0;1� such that for all z :¼ ðl; x; yÞ 2 Bð�zÞ and all h 2 ½0; �h� wehave l > 0 and

hðzþ hDzÞ 6 ½1� rð1� cl� gÞh�hðzÞ; ð2:9Þ

where g 2 ð0;1Þ satisfies g < 1� cl.

Proof. Since rHð�zÞ is nonsingular and �l > 0, there exists a close neighborhood Bð�zÞ of �z such that for any z 2 Bð�zÞ we havel > 0 and that rHðzÞ is invertible. For any z 2 Bð�zÞ, let Dz :¼ ðDl;Dx;DyÞ be a unique solution of the following equation:

rHðzÞðDzÞ ¼ �HðzÞ þlbðzÞ

r

� �; ð2:10Þ

where the residual vector r satisfies krk 6 gjjjHðzÞjjj. From (2.10), for any z 2 Bð�zÞ and all h 2 ½0;1�, we have

lþ hDl ¼ ð1� hÞlþ hlbðzÞ > 0: ð2:11Þ

When hðzÞ > 1; bðzÞ ¼ c < chðzÞ1=2 ¼ cjjjHðzÞjjj, while hðzÞ 6 1; bðzÞ ¼ chðzÞ 6 chðzÞ1=2 ¼ cjjjHðzÞjjj, thus

bðzÞ 6 cjjjHðzÞjjj ð2:12Þ

always holds for any Rþþ �S�S. For any h 2 ð0;1�, let

rðhÞ :¼ Hðzþ hDzÞ � HðzÞ � hrHðzÞDz; ð2:13Þ

then by (2.10) and (2.12), we have

jjjHðzþ hDzÞjjj 6 ð1� hÞjjjHðzÞjjj þ hlkbðzÞk þ hkrk þ jjjrðhÞjjj6 ð1� hÞjjjHðzÞjjj þ hlcjjjHðzÞjjj þ hgjjjHðzÞjjj þ jjjrðhÞjjj6 ½1� rð1� cl� gÞ�jjjHðzÞjjj þ jjjrðhÞjjj: ð2:14Þ

From Lemma 2.1 and (2.11), we know that HðzÞ is continuous differentiable at z 2 Bð�zÞ, which implies jjjrðhÞjjj ¼ oðhÞ. There-fore it follows (2.14) that there exists an �h 2 ð0;1� such that

jjjHðzþ hDzÞjjj 6 ½1� rð1� cl� gÞh�jjjHðzÞjjj ð2:15Þ

holds for all h 2 ð0; �h� and all z 2 Bð�zÞ. Thus, by (2.15), for all h 2 ð0; �h� and all z 2 Bð�zÞ we have

2524 S.-P. Rui, C.-X. Xu / Applied Mathematics and Computation 215 (2009) 2521–2527

hðzþ hDzÞ 6 ½1� rð1� cl� gÞh�2hðzÞ 6 ½1� rð1� cl� gÞh�hðzÞ:

This completes our proof. h

Lemma 2.4. Assume that zk 2NðlÞ and zkþ1 is generated by Algorithm 2.1, then zkþ1 2NðlÞ.

Proof. From l0 ¼ l and the fact bðzkÞ < 1, we have z0 2NðlÞ. If hðzkÞ > 1, then bðzkÞ ¼ c minf1;hðzkÞg ¼ c. Since zk 2NðlÞand bðzÞ 6 c for all z 2 Rþþ �S�S, we have

lbðzkþ1Þ � lkþ1 ¼ lbðzkþ1Þ � ½ð1� hkÞlk þ hklbðzkÞ� 6 lc� ð1� hkÞlk � hklbðzkÞ

6 lc� ð1� hkÞlbðzkÞ � hklbðzkÞ ¼ lðc� bðzkÞÞ ¼ 0: ð2:16Þ

If hðzkÞ 6 1, then bðzkÞ ¼ chðzkÞ, so that by (2.8), hðzk þ hkDzkÞ 6 1, which implies bðzk þ hkDzkÞ ¼ chðzk þ hkDzkÞ. Hence, byinequality (2.8) and zk 2NðlÞ, we have

lbðzkþ1Þ � lkþ1 ¼ lchðzk þ hkDzkÞ � ½ð1� hkÞlk þ hklbðzkÞ�

6 lc½1� rð1� cl� gkÞhk�hðzkÞ � ð1� hkÞlbðzkÞ � hklbðzkÞ

¼ lc½1� rð1� cl� gkÞhk�hðzkÞ � lbðzkÞ ¼ lc½1� rð1� cl� gkÞhk�hðzkÞ � lchðzkÞ

¼ �lcrð1� cl� gkÞhkhðzkÞ 6 0: ð2:17Þ

Combining (2.16) and (2.17), this completes the proof. h

We can get the following result from Lemmas 2.2–2.4.

Theorem 2.1. If F is a continuously differentiable monotone function, then Algorithm 2.1 is well-defined.

3. Convergence analysis

In this section, we gives the global convergence and local superlinear convergence of Algorithm 2.1. To discuss the globalconvergence of Algorithm 2.1, we need the following assumption.

Assumption 3.1. The neighborhood NðlÞ given by (2.7) is bounded.

The Assumption 3.1 is also a fundamental assumption for the non-interior continuation methods for complementarityproblems [6,8,19,21,22]. The following theorem gives the global convergence of Algorithm 2.1.

Theorem 3.1. Assume that F is a continuous differentiable monotone function and the sequence fzkg is generated by Algorithm 2.1. If Assumption 3.1 holds, then any accumulation point of the sequence is a solution of Hðl; x; yÞ ¼ 0.

Proof. By Assumption 3.1, fzkg is bounded. Let z� :¼ ðl�x�y�Þ is an accumulation point of fzkg. By taking a subsequenceof fzkg if necessary that converges to fz�g, for sake of simplicity, we assume that fzkg converges to fz�g. From thedesign of Algorithm 2.1, the sequence fhðzkÞg is monotonically decreasing and bounded from below by zero. Hence,we assume that fhðzkÞg converges to a nonnegative number h�. If h� ¼ 0, then it follows from the continuity of functionhð�Þ that hðz�Þ ¼ 0. We obtain the desired result. In what follows we assume h� > 0, and then deduce to a contradiction.By hðz�Þ > 0 and lkþ1 ¼ ð1� hkÞlk þ bc minf1;hðzkÞg, we have l� 2 Rþþ, then z� 2NðblÞ and rHðz�Þ is nonsingular.Hence, from Lemma 2.3, there exists a closed neighborhood Bðz�Þ of z� and a positive number �h 2 ð0;1�, such thatfor any z 2 Bðz�Þ and all h 2 ½0; �h�, we have l > 0; rHðzÞ is nonsingular and inequality (2.9) holds. Since fzkg convergesto z�, then zk 2 Bðz�Þ for all sufficiently large k. Hence, there is a nonnegative integer number l such that, whendl 2 ð0; �h�,

hðzk þ dlDzkÞ 6 ½1� rð1� cl� gÞdl�hðzkÞ ð3:1Þ

holds for all sufficiently large k. By the choice of stepsize hk in Algorithm 2.1, clearly, hk P dl for all sufficiently large k. It fol-lows that

hðzkþ1Þ 6 ½1� rð1� cl� gÞdl�hðzkÞ ð3:2Þ

holds for all sufficiently large k. Notice that dl is independent to k. Taking the limit k!1 in the two sides of inequality (3.2),it follows that

h� 6 ½1� rð1� cl� gÞdl�h� ) h� 6 0:

This contradicts h� > 0. So, we complete our proof. h

In what follows, we study the local convergence of Algorithm 2.1. For our analysis, we need the following assumption:

S.-P. Rui, C.-X. Xu / Applied Mathematics and Computation 215 (2009) 2521–2527 2525

Assumption 3.2. Let z� ¼ ð0; x�y�Þ be an accumulation point of the sequence fzkg generated by Algorithm 2.1 that satisfiesthe following conditions: (a) ðstrict complementarityÞ x� þ y� � 0; (b) ðnondegeneracyÞ the equations x�v þ uy� ¼ 0 andrFðx�Þu ¼ v have ðu;vÞ ¼ 0 as the only solution.

Assumption 3.2(a) is rather standard, and Assumption 3.2(b) was introduced by Kojima et al. [4]. Assumption 3.2 impliesthat the linear rHðz�Þ is invertible, the proof is similar to Remark 4.1 in [5]. In order to get our main locally superlinearlyconvergent result, we also need the following lemmas.

Lemma 3.1. Let z� ¼ ð0; x�y�Þ be an accumulation point of the sequence fzkg generated by Algorithm 2.1 . If Assumption 3.2 holds,then we have

hðzkÞ ¼ Oðjjjzk � z�jjj2Þ; ðk!1Þ:

Proof. From Lemma 2.2, we have that rHðzÞ is nonsingular for all z 2 Rþþ �S�S. By Assumption 3.2, rHðz�Þ is nonsin-gular. Then, for all sufficiently large k, it follows from Lemma 2.1 and Hðz�Þ ¼ 0 that

jjjHðzkÞjjj ¼ jjjHðzkÞ � Hðz�Þjjj ¼ jjjZ 1

0rHðz� þ aðzk � z�ÞÞðzk � z�Þdajjj 6

Z 1

0jjjrHðz� þ aðzk � z�ÞÞðzk � z�Þjjjda

¼ Oðjjjzk � z�jjjÞ:

This completes our proof. h

Theorem 3.2. Suppose F is a continuously differentiable monotone function, z� be an accumulation point of the sequence fzkg gen-erated by Algorithm 2.1 . If Assumption 3.2 holds, then sequence fzkg linearly converges to z�.

Proof. Lemma 2.2 implies thatrHðzkÞ is nonsingular for all zk 2NðlÞ, then there exists a positive real number x, such thatfor any zk 2NðlÞ,

jjjrH�1ðzkÞjjj 6 x: ð3:3Þ

By Lemma 2.2, (2.6), (3.3) and bðzkÞ 6 cffiffiffiffiffiffiffiffiffiffiffihðzkÞ

p¼ cjjjHðzkÞjjj, for all sufficiently large k, we have

jjjDzkjjj ¼ jjjrH�1ðzkÞ½HðzkÞ þ ðlbðzkÞ; rkÞ�jjj 6 x½jjjHðzkÞjjj þ lcjjjHðzkÞjjj þ krkk�

6 x½jjjHðzkÞjjj þ lcjjjHðzkÞjjj þ ð1� lcÞjjjHðzkÞjjj� ¼ OðjjjHðzkÞjjjÞ ¼ OðffiffiffiffiffiffiffiffiffiffiffihðzkÞ

qÞ: ð3:4Þ

Then, by (3.4) and Lemma 3.1, for all sufficiently large k, we have

jjjðzkþ1 � z�Þjjj ¼ jjjðzkþ1 � zkÞ þ ðzk � z�Þjjj 6 hkjjjDzkjjj þ jjjðzk � z�Þjjj ¼ OffiffiffiffiffiffiffiffiffiffiffihðzkÞ

q� �þ jjjðzk � z�Þjjj

¼ Oðjjjðzk � z�ÞjjjÞ; ðk!1Þ:

This indicates the sequence fzkg generated by Algorithm 2.1 at least linearly converges to fz�g. h

Lemma 3.2. Suppose that F is a continuously differentiable monotone function, rF is Lipschitz continuous, Assumptions 3.1 and3.2 hold. If the residual vector sequence rk satisfies krkk ¼ oðjjjHðzkÞjjjÞ, then

zkþ1 ¼ zk þ Dzk

holds for all sufficiently large k, i.e., stepsize hk ¼ 1 always satisfies inequality (2.8) for every sufficiently large k.

Proof. We assume that zk is generated by Algorithm 2.1 and converges to z�. By Theorem 3.1, z� is a solution of SDCP (1.1).From Assumption 3.2 it follow thatr/ðzÞ is Lipschitz continuous in Nðz�eÞ for some small e > 0. Since z� is an accumulationpoint of the sequence fzkg, so that fzkg 2Nðz�eÞ for any sufficiently large k. Thus, there exists L > 0 such that

jjjHðzk þ DzkÞ � HðzkÞ � rHðzkÞDzkjjj 6 LjjjDzkjjj2 ð3:5Þ

holds for any sufficiently large k. If jjjHðzkÞjjj ¼ 0, then zkþ1 ¼ zk þ Dzk. Thus, we may assume in the following that jjjHðzkÞjjj–0for any sufficiently large k. It follows from (2.6) and (3.3) that

jjjDzkjjj 6 x½jjjHðzkÞjjj þ lcjjjHðzkÞjjj þ krkk�: ð3:6Þ

Hence, from (2.6), (3.5) and (3.6) we have

jjjHðzk þ DzkÞjjj ¼ jjjHðzk þ DzkÞ � HðzkÞ � rHðzkÞDzk þ ðlbðzkÞ; rkÞjjj 6 LjjjDzkjjj2 þ lcjjjHðzkÞjjj þ krkk

6 Lx2ðjjjHðzkÞjjj þ lcjjjHðzkÞjjj þ krkkÞ2 þ cjjjHðzkÞjjj2 þ krkk:

2526 S.-P. Rui, C.-X. Xu / Applied Mathematics and Computation 215 (2009) 2521–2527

As jjjHðzkÞjjj ! 0ðk!1Þ and krkk ¼ oðjjjHðzkÞjjjÞ, then we have jjjHðzk þ DzkÞjjj ¼ oðjjjHðzkÞjjjÞ, that is

hðzk þ DzkÞ ¼ oðhðzkÞÞ: ð3:7Þ

Thus, for any e 2 ð0;1Þ, in particular, taking e ¼ 1� rð1� cl� gkÞ; jjjhðzk þ DzkÞjjj 6 ½1� rð1� cl� gkÞ�jjjhðzkÞjjj holds for allsufficiently large k, i.e., zkþ1 ¼ zk þ Dzk will be accepted as the next iterate point of Algorithm 2.1. h

Theorem 3.3. Suppose that F is a continuously differentiable monotone function, fzkg is generated by Algorithm 2.1 and convergesto z�. If rF is Lipschitz continuous and Assumption 3.2 holds, the residual vector sequence rk satisfies krkk ¼ oðjjjHðzkÞjjjÞ, then thesequence fzkg superlinearly converges to z�, that is,

jjjðzkþ1 � z�Þjjj ¼ ojjjðzk � z�Þjjj

and

lkþ1 ¼ oðlkÞ:

Proof. First, since HðzkÞ is continuously differentiable for all zk 2NðlÞ (where NðlÞ is defined by (2.7)) and rHðzkÞ is non-singular, it yields that for all sufficiently large k,

jjjHðzkÞ � Hðz�Þ � rHðzkÞðzk � z�Þjjj ¼ oðjjjzk � z�jjjÞ: ð3:8Þ

Then by Lemmas 3.1, 3.2, (3.3) and (3.8) we have

jjjzkþ1 � z�jjj ¼ jjjzk þ Dzk � z�jjj ¼ jjjzk � z� � rH�1ðzkÞ½HðzkÞ � ðlbðzkÞ; rkÞ�jjj6 x½jjjHðzkÞ � H0ðz�Þ � rHðzkÞðzk � z�Þjjj þ lhðzkÞ þ krkk� 6 x½oðjjjzk � z�jjjÞ� ¼ oðjjjzk � z�jjjÞ: ð3:9Þ

Next, since zkþ1 ¼ zk þ Dzk from Lemma 3.2, it follows that for all sufficiently large k,

lkþ1 ¼ lk þ Dlk ¼ lbðzkÞ ¼ lchðzkÞ:

Combing with (3.7), we obtain

lkþ1=lk ¼ hðzkÞ=hðzk�1Þ ! 0; ðk!1Þ:

This implies that we complete the proof. h

4. Conclusions

We have proposed an inexact algorithm for large-scale monotone SDCP under the framework of non-interior continuationmethod. The superlinearly convergent ratio was showed for this method. Such an inexact method for solving the large-scalemonotone SDCP offers a trade-off between the accuracy of solving the subproblems and the amount of work for solvingthem. Furthermore, under suitable assumption, if the residual vector r is further restricted, such as krk ¼ OðjjjHðzÞjjj2Þ, weactually may obtain a quadratic rate of convergence for our algorithm.

References

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