The Scientific World Journal
Volume 2014 (2014), Article ID 307823, 6 pages
http://dx.doi.org/10.1155/2014/307823

Research Article

Levenberg-Marquardt Method for the Eigenvalue Complementarity Problem

Yuan-yuan Chen1,2 and Yan Gao¹

¹School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China
²College of Mathematics, Qingdao University, Qingdao 266071, China

Received 22 June 2014; Revised 28 August 2014; Accepted 29 August 2014; Published 30 October 2014

Academic Editor: Pu-yan Nie

Copyright © 2014 Yuan-yuan Chen and Yan Gao. 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.

The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, including nonsmooth or semismooth Newton type methods. In this paper, we reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them. Under mild assumptions, the method is proved globally convergent. Finally, some numerical results and the extensions of the method are also given. The numerical experiments highlight the efficiency of the method.

1. Introduction

Eigenvalue complementarity problem (EiCP) is proposed in the study of the problems in mechanics, engineering, and economics. The EiCP is also called cone-constrained eigenvalue problem in [1–4]. The EiCP is to find a solution including a scalar and a nonzero vector satisfying a complementarity constraint on a closed convex cone. The EiCP can be reformulated to be a special complementarity problem or a mathematical programming optimization problem with complementarity constraints and can use nonsmooth or semismooth Newton type method to solve it, such as [5–7]. The Levenberg-Marquardt method is one of the widely used methods in solving optimization problems (see, for instance, [8–15]). Use a trust region strategy to replace the line search, the Levenberg-Marquardt method is widely considered to be the progenitor of the trust region method approach for general unconstrained or constrained optimization problems. The use of a trust region avoids the weaknesses of Gauss Newton method, that is, its behavior when the Jacobian is rank deficient or nearly so rank deficient. On the other hand, we reformulate the EiCP as a system of continuously differentiable equations that is one of the most interesting themes. The advantage of the reformulation is that we solve the equations with continuously differentiable functions for which there are rich powerful solution methods and theory analysis, including the powerful Levenberg-Marquardt method. So, in this paper, we give the Levenberg-Marquardt method to solve the EiCP. The EiCP, which we will consider, is the following problem. Given the matrix and the matrix , which are positive definite matrix, then we consider to find a scalar and a vector , such that

This paper is organized as follows. In Section 2, we give some background definitions and known properties. And we also give the Levenberg-Marquardt method for the EiCP. The global convergence analysis and some discussions of the Levenberg-Marquardt method is also given. In Sections 3 and 4, we give some numerical results and some extensions of the method.

Throughout the paper, denotes a real matrix of order and denotes a real matrix of order . and .

2. Preliminaries

In this section, firstly, we reformulate (1) as a system of continuously differentiable equations and give some preliminaries used in the following. Then, we propose the Levenberg-Marquardt method for the EiCP.

As we all know, (1) can be rewritten as

where is a continuously differentiable function and the matrix . Using the nonlinear complementarity problem (NCP) function , which satisfied the following basic property

By the property, (2) can be recasted as the following system of equations:

Then, solves the EiCP (1) if and only if solves (4). Define a merit function for (4) as

We use a special NCP function named Fischer-Burmeister function, defined as

We know that the favorable property of is that is a continuously differentiable function on the whole space, although is not a continuously differentiable function in general (see, for example, [16]). Thus, we give the system of continuously differentiable equations for (1) as the following equations:

where , which is a continuously differentiable function. In what follows, we will give the Levenberg-Marquardt method. Denote

The least-squares formulation of (7) is the following unconstrained optimization problem:

Now, we give the Levenberg-Marquardt method for solving (1). The global convergence result of the method is also given.

Method 1 (the Levenberg-Marquardt method for the EiCP). Given , , , , . , , , , and .

Step 1. If , then stop. Otherwise, compute by

Step 2. If , let ; otherwise, is computed by (10). Then, find the smallest nonnegative integer such that

Set .

Step 3. Let and , and go to Step 1.

Now, we give the global convergence of Method 1.

Theorem 1. Suppose that , generated by Method 1. Then, each accumulation point of the sequence is a stationary point of .

Proof. Suppose that , is a subsequence of , and . When there are infinitely many such that , by Proposition  1.16 in [17], we get the assertion. In the following, we assume that if is a convergent subsequence of , then is always computed by (10). We assume that for every convergent subsequence for which

we have

In the following, we also assume that . Suppose that is not a stationary point of . By (10), we have

therefore, we have

Obviously, the denominator in the above inequality is nonzero; otherwise, we have . Then, the algorithm has stopped. On the other hand, we know that there exists a constant such that

moreover,

By and the fact that the gradient is bounded on the convergent sequence , we get (13). We next prove (14). If (14) is not satisfied, there exists a subsequence of :

This implies that . From (18), we know that

which contradicts with (12). Thus, (14) holds. So, according to the definition given in [17], the sequence is uniformly gradient related to . We complete the proof.

Remark 2. In Method 1, we can also use some other line search, such as the nonmonotone line search. The line search is to find the smallest nonnegative integer such that

where , and is a integer.

Remark 3. In Method 1, we can also use the following equation to compute in Step 1. We can find an approximate solution of the equation

where is the residuals and satisfies

where for every .

3. Numerical Results

We give some numerical experiments for the method. And we compare Method 1 with the scaling and projection algorithm (denoted by SPA in [18]). The numerical results indicate that Method 1 works quite well in practice. We consider the eigenvalue complementarity problems, which are all taken form [3, 18]. All codes for the method are finished in MATLAB. The parameters used in the method are chosen as , , , and .

Example 4. We consider

where

By [3], we know that Example 4 has three eigenvalues. Now, we consider random initial points to compute Example 4 by Method 1. Numerical results for Example 4 by Method 1 and SPA method are presented in Table 1.

Table 1

Method 1

SPA

Table 1 shows that Method 1 are able to detect all the solutions for the small size matrix. But the SPA method can only detect 2 solutions from [3].

Example 5. We consider

where

From [3], we also know that Example 5 have 9 eigenvalues. Now, we consider random initial points to compute the above Example 5 by Method 1. Numerical results for Example 5 by Method 1 and the SPA method are presented in Table 2.

Table 2

Method 1

SPA

Table 2 shows that Method 1 is able to detect all the solutions. But, the SPA method can only detect 4 Pareto eigenvalues from the analysis of [3].

Example 6. We consider

where

From [3], we also know that Example 6 have 23 eigenvalues. Now, we consider random initial points to compute Example 6 by Method 1. Numerical results for Example 6 by Method 1 and the SPA method are given in Table 3.

Table 3

Method 1

SPA

26.28209

100

26.41489

158

28.71143

29.13415

32.60775

32.86388

37.57690

41.01635

46.46811

49.14424

66.96950

77.42278

77.45752

99.42333

99.99801

107.49890

127.38892

148.53897

157.99623

197.17149

204.57942

226.27891

231.92132

Table 3 shows that Method 1 is able to detect all 23 solutions. But, the SPA method can only detect 2 Pareto eigenvalues from [3]. The numerical results indicate that Method 1 works quite well for the big size EiCP in practice.

Discussion. In this section, we study the numerical behaviors of Method 1 for solving the Pareto eigenvalue problem. The EiCP problem is very useful in studying the optimization problems arising in many areas of the applied mathematics and mechanics. By using the F-B function, we reformulate the EiCP as a system of continuously differentiable equations. Then, we use the Levenberg-Marquardt method to solve it. From the above numerical results, we know that Method 1 is very effective for small size and big size EiCP problems.

4. Extensions

4.1. Bieigenvalue Complementarity Problems (BECP)

We also can use Method 1 to solve the bieigenvalue complementarity problems (denoted by BECP). The BECP is to find and such that

where , , , and .

Let , , , , and . We can write the above bieigenvalue complementarity problems as , , , , , , , and . By using F-B function, similar to rewriting the EiCP, we can rewrite the above bieigenvalue complementarity problems as the following equations:

Let

Then, we can give the system of continuously differentiable equations for the bieigenvalue complementarity problems as the following continuously differentiable equations:

In the following numerical test, the parameters used in the method are also chosen as , , , and .

Example 7. We consider the BECP, where , ,

We use Method 1 to solve the Example 7. Results for Example 7 with random initial point are given in Table 4.

Table 4

0.95803
0.96145

0.96205
0.97990

0.95574
0.94739

0.96069
0.93502

0.94582
1.04493

0.96200
1.03264

0.95517
1.06445

0.95744
1.05645

4.2. The Paretian Version

We consider using Method 1 to compute the following Paretian version:

where standing for the pencil associated to a finite collection of real matrices of order and

Example 8. We consider the quadratic pencil model, where

We use Method 1 to compute Example 8. We present the results of Example 8 with random initial point in Table 5.

Table 5

0.25315

5. Conclusion

In this paper, we reformulate the EiCP as a system of continuously differentiable equations and use the Levenberg-Marquardt method to solve them. The numerical experiments show that our method is a promising method for solving the EiCP. The numerical experiments of the extensions confirm the efficiency of our method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Science Foundation of China (11171221, 11101231), Shanghai Leading Academic Discipline (XTKX 2012), and Innovation Program of Shanghai Municipal Education Commission (14YZ094).

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