Journal of Applied Mathematics

Volume 2013 (2013), Article ID 571927, 10 pages

http://dx.doi.org/10.1155/2013/571927

Research Article

A Two-Parametric Class of Merit Functions for the Second-Order Cone Complementarity Problem

Xiaoni Chi,1,2 Zhongping Wan,¹ and Zijun Hao1,3

¹School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

²School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China

³School of Information and Calculating Science, North University for Ethnics, Yinchuan 750021, China

Received 18 February 2013; Revised 16 May 2013; Accepted 20 May 2013

Academic Editor: Zhongxiao Jia

Copyright © 2013 Xiaoni Chi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We propose a two-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) based on the one-parametric class of complementarity functions. By the new class of merit functions, the SOCCP can be reformulated as an unconstrained minimization problem. The new class of merit functions is shown to possess some favorable properties. In particular, it provides a global error bound if and have the joint uniform Cartesian -property. And it has bounded level sets under a weaker condition than the most available conditions. Some preliminary numerical results for solving the SOCCPs show the effectiveness of the merit function method via the new class of merit functions.

1. Introduction

We consider the following second-order cone complementarity problem (SOCCP) of finding such that

where is the Euclidean inner product and and are continuously differentiable mappings. Here is the Cartesian product of second-order cones (SOC); that is, with , , and the -dimensional SOC defined by

with denoting the Euclidean norm.

Recently great attention has been paid to the SOCCP, since it has a variety of engineering and management applications, such as filter design, antenna array weight design, truss design, and grasping force optimization in robotics [1, 2]. Furthermore, the SOCCP contains a wide class of problems, such as nonlinear complementarity problems (NCP) and second-order cone programming (SOCP) [3, 4]. For example, the SOCCP with and for any is the NCP, and the KKT conditions for the SOCP reduce to the SOCCP.

There have been various methods for solving SOCCPs [5], such as interior point methods [6–8], (noninterior continuation) smoothing Newton methods [4, 9–12], and smoothing-regularization methods [13]. Recently, there is an alternative approach [14, 15] based on reformulating the SOCCP as an unconstrained smooth minimization problem. In that approach, it aims to find a smooth function such that

Such a is called a merit function for the SOCCP. Thus the SOCCP is equivalent to the following unconstrained smooth (global) minimization problem:

A popular choice of is the Fischer-Burmeister (FB) merit function

where is the vector-valued FB function defined by

with denoting the Jordan product between and itself and being a vector such that . The function is shown to be a merit function for the SOCCP in [14].

In this paper, we consider the two-parametric class of merit functions defined by

where and are given, respectively, by

with denoting the metric projection on the second-order cone , and . Here is the one-parametric class of SOC complementarity functions [16] defined by

where is an arbitrary but fixed parameter. When , reduces to the vector-valued FB function given by (6), and as , it becomes a multiple of the vector-valued residual function

Thus, the one-parametric class of vector-valued functions (10) covers two popular second-order cone complementarity functions. Hence the two-parametric class of merit functions defined as (7)–(10) includes a broad class of merit functions.

We will show that the SOCCP can be reformulated as the following unconstrained smooth (global) minimization problem:

If and , the function in (12) induced by the new class of merit functions reduces to [17]

with given as (8). It has been shown that provides a global error bound if and are jointly strongly monotone, and it has bounded level sets if and are jointly monotone and a strictly feasible solution exists [17]. In contrast, the merit function lacks these properties.

Motivated by these works, we aim to study the two-parametric class of merit functions for the SOCCP defined as (7)–(10) and its favorable properties in this paper. We also prove that the class of merit functions provides a global error bound if and have the joint uniform Cartesian -property, which will play an important role in analyzing the convergence rate of some iterative methods for solving the SOCCP. And it has bounded level sets under a rather weak condition, which ensures that the sequence generated by a descent method has at least one accumulation point.

The organization of this paper is as follows. In Section 2, we review some preliminaries including the Euclidean Jordan algebra associated with SOC and some results about the one-parametric class of SOC complementarity functions. In Section 3, based on the one-parametric class of SOC complementarity functions, we propose a two-parametric class of merit functions for the second-order cone complementarity problem (SOCCP), which is shown to possess some favorable properties. In Section 4, we show that the class of merit functions provides a global error bound if and have the joint uniform Cartesian -property, and it has bounded level sets under a rather weak condition. Some preliminary numerical results are reported in Section 5. And we close this paper with some conclusions in Section 6.

In what follows, we denote the nonnegative orthant of by . We use the symbol to denote the Euclidean norm defined by for a vector or the corresponding induced matrix norm. For simplicity, we often use for the column vector . For the SOC , , and mean the topological interior and the boundary of , respectively.

2. Preliminaries

In this section, we recall some preliminaries, which include Euclidean Jordan algebra [3, 18] associated with the SOC and some results used in the subsequent analysis.

Without loss of generality, we may assume that and in Sections 2 and 3.

First, we recall the Euclidean Jordan algebra associated with the SOC and some useful definitions. The Euclidean Jordan algebra for the SOC is the algebra defined by

with being its unit element. Given an element , we define

where represents the identity matrix. It is easy to verify that for any . Moreover, is symmetric positive definite (and hence invertible) if and only if .

Now we give the spectral factorization of vectors in associated with the SOC . Let . Then can be decomposed as

where and are the spectral values and the associated spectral vectors of given by

for , with any such that . It is obvious that . By the spectral factorization, a scalar function can be extended to a function for the SOC. For any , we define

Since both eigenvalues of any are nonnegative, we define

Lemma 1 (see [14]). Let be any closed convex cone in . For each , let and denote the nearest point (in the Euclidean norm) projection of onto and , respectively. The following results hold.

(i)For any , we have and .(ii)For any and , we have .

Lemma 2 (see [19]). Let and . Then we have

The following results, describing the special properties of the function given as (10), will play an important role in the subsequent analysis.

Lemma 3 (see [16]). For any , , if , then

If, in addition, , then , and furthermore,

Lemma 4 (see [20]). For any , let be defined as in (10). Then,

3. A Two-Parametric Class of Merit Functions

In this section, we study the two-parametric class of merit functions given by (7)–(10). As we will see, has some favorable properties. The most important property is that the SOCCP can be reformulated as the global minimization of the function given as (12). Moreover, the function provides a global error bound and bounded level sets under weak conditions, which will be shown in the next section.

Proposition 5. Let be given by (9). Then,

Proof. Suppose , , and . Thus by Proposition 3.1 [16], we have and therefore , . Conversely, we assume and . Then implies . From (10), we obtain

Squaring both sides yields

Taking the trace of both sides and using the fact , we have

Since and , we obtain

and thus the right hand side of (27) is nonnegative. Then by the assumption , we have . This together with (27) implies . Therefore, it follows from Proposition 3.1 [16] that and .

Proposition 6. The function given by (9) is differentiable at any . Moreover, ; if , then

if and , then , and

Proof

Case 1. If , then for any , , let be the spectral values of and let be the corresponding spectral vectors. Then,

It follows from the definition of spectral value that

Combining (31) and (32) together with Lemma 4 yields that

This shows that is differentiable at with .

Case 2. If , let be defined by for any . By the proof of Proposition 3.2 [17], is continuously differentiable and . Since and is differentiable at any satisfying by Proposition 3.2 [16], is differentiable in this case and

Case 3. If and , it follows from Lemma 3 that

In this case, direct calculations together with Lemma 3 yield

Thus, the bigger spectral value of and its corresponding spectral vector are given as

where

By the spectral factorization, we have

Therefore, we prove the differentiability of in this case by considering the following three subcases.

(i) If , then , where are given by (37). Then we have

It is obvious that is differentiable in this case. Moreover, by Lemma 3, we have

Therefore, the derivative of with respect to is

and the gradient of with respect to is

Then it follows from (37), (42), and (43) that can be rewritten as

Similarly, we can show that

(ii) If , we have and thus . Then by Proposition 3.2 [16], the gradient of is

If there exists such that and , it follows from (44)–(46) that

Therefore, is differentiable in this subcase.

(iii) If , we have and thus . Then it is obvious that the gradient of is

If there exists such that and , it follows from (44), (45), and (48) that

Therefore, is differentiable in this subcase.

Proposition 7. Let be given by (9). For any , , we have

and the equality in (51) holds whenever .

Proof. By following the proof of Lemma 4.1 [16] and using Proposition 6, we can show that the desired results hold.

Proposition 8. Let be given by (7)–(10) and (12). Then, the following results hold.

(i)For all , we have and if and only if solves the SOCCP.(ii)If and are differentiable, the function is differentiable with

Proof. (i) It is obvious that for all . Now we prove if and only if solves the SOCCP. Suppose . Then we have , and which implies and therefore . By Proposition 5, the last relation together with yields , , . Therefore, solves the SOCCP. On the other hand, suppose that solves the SOCCP. Then ,  ,  , which are equivalent to , , [4]. By Proposition 5, (7), (8), and (12), we have , , and therefore .

(ii) From Lemma 3.1 [19], we have that the function is differentiable for all with and . Then, by the chain rule and direct calculations, the result follows.

4. Error Bound and Bounded Level Sets

By Proposition 8, we see that the SOCCP is equivalent to the global minimization of the function . In this section, we show that the function provides a global error bound for the solution of the SOCCP and has bounded level sets, under rather weak conditions.

In this section, we consider the general case that is the Cartesian product of SOCs; that is, with , . Thus, we obtain

and therefore the results in Sections 2 and 3 can be easily extended to the general case.

First, we discuss under what condition the function provides a global error bound for the solution of the SOCCP. To this end, we need the concepts of Cartesian -properties introduced in [21] for a nonlinear transformation, which are natural extensions of the -properties on Cartesian products in established by Facchinei and Pang [22]. Recently, the Cartesian -properties are extended to the context of general Euclidean Jordan algebra associated with symmetric cones [20].

Definition 9. The mappings and are said to have

(i)the joint uniform Cartesian -property if there exists a constant such that, for every , there is an index such that

(ii)the joint Cartesian -property if, for every with , there is an index such that

Now we show that the function provides a global error bound for the solution of the SOCCP if and have the joint uniform Cartesian -property.

Proposition 10. Let be given by (7)–(10) and (12). Suppose that and have the joint uniform Cartesian -property and the SOCCP has a solution . Then there exists a constant such that, for any ,

Proof. Since and have the joint uniform Cartesian -property, there exists a constant such that, for any , there is an index such that

where the second inequality is due to Lemma 1(ii) and the third inequality follows from Lemma 2. Setting , we obtain

By (7), (8), and (12), we have

Moreover, we obtain from Lemma 4 that

Combining (58), (59), and (60) yields the desired result.

To guarantee the boundedness of the level sets

for any , we give the following condition.

Condition 11. For any sequence such that

there holds that

Proposition 12. If the mappings and satisfy Condition 11, then the level sets of for any are bounded.

Proof. On the contrary, we assume that there exists an unbounded sequence for some . Thus and for all . Then by Lemma 4, we have for all that

which implies and . Therefore, from Condition 11, there is such that . It follows from (7), (8), and (12) that

and hence . This contradicts the fact that .

It should be noted that Condition 11 is rather weak to guarantee the boundedness of level sets of . As far as we know, the weakest condition available to ensure the boundedness of level sets is the following condition given by [20].

Condition 13 (see [20]). For any sequence such that

there holds that

It is obvious that for any , and therefore Condition 13 implies Condition 11. It has been shown that the symmetric cone complementarity problem (SCCP) with the jointly monotone mappings and a strictly feasible point, or the SCCP with joint Cartesian property [23], all imply Condition 13 [20]. Hence they all implies Condition 11, since the SCCP includes the SOCCP. Therefore, Condition 11 is a weaker condition than the most available conditions to guarantee the boundedness of level sets.

5. Numerical Results

In this section, we employ the merit function method based on the unconstrained minimization reformulation (12) to solve the SOCCPs (1). All the experiments were performed on a desktop computer with Intel Pentium Dual T2390 CPU 1.86 GHz and 1.00 GB memory. The operating system was Windows XP and the implementations were done in MATLAB 7.0.1.

We adopt the L-BFGS method [24], a limited-memory quasi-Newton method, with 5 limited-memory vector-updates to solve the unconstrained minimization reformulation (12) where the two-parametric class of merit functions is given as (7)–(10). For the scaling matrix in the L-BFGS, we adopt as recommended by [25], where

In the L-BFGS, we revert to the steepest descent direction whenever . In addition, we use the nonmonotone line search [26] to seek a suitable steplength. In detail, we compute the smallest nonnegative integer such that

where denotes the direction in the th iteration generated by the L-BFGS, and are parameters in (0,1), and is given by

where, for a given nonnegative integer and , we set

Throughout the numerical experiments, we choose the following parameters:

The algorithm is stopped whenever the number of function evaluations for is over 10000 or as the stopping criterion.

The test problems are the randomly generated linear SOCCPs (1), where

with and . In detail, we generate a random matrix and a random vector , and then let . Since the matrix is semidefinite positive, the generated problems (1) are the monotone linear SOCCPs. In the tables of test results, denotes the size of problems; NF denotes the (average) number of iterations; CPU(s) denotes the (average) CPU time in seconds; and Gap denotes the (average) value of when the algorithm terminates.

We solve the linear SOCCPs of different dimensions with size from 50 to 1000 and . The random problems of each size are generated 10 times, and the test results with different parameters and are listed in Tables 1, 2, and 3. From the results of these tables, we give several observations.

(i)All the random problems have been solved in very short CPU time.(ii)The problem size slightly affects the number of iterations.(iii)For the same dimension of linear SOCCPs, choices of different parameters and generally do not affect the number of iterations and the CPU time.

Table 1: Numerical results for SOCCPs with .

NF
CPU (s)
Gap

50
155.1
0.063

100
162.0
0.078

150
166.0
0.171

200
168.1
0.297

250
170.3
0.563

300
172.2
0.953

350
174.2
1.453

400
175.0
2.094

450
176.0
2.875

500
177.0
3.766

550
178.0
4.984

600
178.2
6.234

650
179.1
7.938

700
180.3
9.687

750
181.6
11.594

800
181.0
14.297

850
182.7
16.625

900
182.6
19.703

950
183.0
22.922

1000
183.2
26.438

Table 2: Numerical results for SOCCPs with .

NF
CPU (s)
Gap

50
156.0
0.062

100
162.1
0.078

150
166.1
0.172

200
169.2
0.313

250
171.6
0.547

300
172.3
0.953

350
174.5
1.469

400
175.6
2.094

450
176.8
2.859

500
177.0
3.781

550
178.0
4.984

600
178.2
6.250

650
179.0
7.891

700
180.5
9.765

750
181.2
11.687

800
181.0
14.266

850
182.1
16.609

900
182.0
19.563

950
183.0
23.000

1000
183.0
26.360

Table 3: Numerical results for SOCCPs with .

NF
CPU (s)
Gap

50
157.0
0.063

100
162.1
0.094

150
166.1
0.172

200
169.0
0.328

250
171.2
0.562

300
172.2
0.969

350
174.3
1.437

400
175.0
2.125

450
176.0
2.875

500
177.3
3.781

550
178.2
4.937

600
178.5
6.266

650
179.6
7.829

700
180.2
9.641

750
181.0
11.625

800
181.0
14.312

850
182.0
16.610

900
182.0
19.578

950
183.0
22.922

1000
183.1
26.390

6. Conclusions

In this paper, we have studied a two-parametric class of merit functions for the second-order cone complementarity problem based on the one-parametric class of complementarity functions. The new proposed class of merit functions includes a broad class of merit functions, since the one-parametric class of complementarity functions is closely related to the famous natural residual function and Fischer-Burmeister function. The new class of merit functions has been shown to possess some favorable properties. In particular, it provides a global error bound if and have the joint uniform Cartesian -property. And it has bounded level sets under a weaker condition than the most available conditions [20, 23]. Some preliminary numerical results for solving the SOCCPs show the effectiveness of the merit function method via the new class of merit functions.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (no. 71171150), China Postdoctoral Science Foundation (no. 2012M511651), and the Excellent Youth Project of Hubei Provincial Department of Education (no. Q20122709), China. The authors are grateful to the editor and the anonymous referees for their valuable comments on this paper. Particulary, the authors thank one of the referees for his helpful suggestions on numerical results, which have greatly improved this paper.

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