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On the preconditioned GAOR method for a linear complementarity problem with an Mmatrix
Journal of Inequalities and Applications volume 2018, Article number: 195 (2018)
Abstract
Recently, based on the Hadjidimos preconditioner, a preconditioned GAOR method was proposed for solving the linear complementarity problem (Liu and Li in East Asian J. Appl. Math. 2:94–107, 2012). In this paper, we propose a new preconditioned GAOR method for solving the linear complementarity problem with an Mmatrix. The convergence of the proposed method is analyzed, and the comparison results are obtained to show it accelerates the convergence of the original GAOR method and the preconditioned GAOR method in (Liu and Li in East Asian J. Appl. Math. 2:94–107, 2012). Numerical examples verify the theoretical analysis.
Introduction
For a given matrix \(A\in \mathbb{R}^{n\times n}\) and a vector \(f\in \mathbb{R}^{n}\), the linear complementarity problem, abbreviated as LCP, consists of finding a vector \(x\in \mathbb{R}^{n}\) such that
Here, the notation “≥” denotes the componentwise defined partial ordering between two vectors and the superscript T denotes the transpose of a vector.
The LCP of the form (1) arises in many scientific computing and engineering applications, for example, the Nash equilibrium point of a bimatrix game, the contact problem, and the free boundary problem for journal bearings; see [5, 16] and the references therein. As is known, LCP (1) possesses a unique solution if and only if \(A\in \mathbb{R}^{n\times n}\) is a Pmatrix, namely a matrix whose all principal submatrices have positive determinants; see [4, 5, 16]. A matrix A is called an Mmatrix if its inverse is nonnegative and all its offdiagonal entries are nonpositive. A positive diagonal Mmatrix is a Pmatrix, and LCP (1) with an Mmatrix has the unique solution [3].
Because of the wide applications, the research on the numerical methods for solving (1) has attracted much attention. There are some iterative methods for obtaining the solution of the LCP, including the projected methods [8, 9, 12], the modulus algorithms [10], and the modulusbased matrix splitting iterative methods [2, 6, 18, 19], see [9] for a survey of the iterative method for LCP (1). We consider the generalized AOR (GAOR) method [8, 12], which is a special case of the projected method, for solving LCP (1) with an Mmatrix. For accelerating the convergence rate of the GAOR method, the preconditioned GAOR method is proposed in [13], based on the preconditioner in [7], for LCP (1) with an Mmatrix. In this paper, a new preconditioner is proposed to accelerate the convergence rate of the GAOR method for solving LCP (1).
The outline of the rest of the paper is as follows. In Sect. 2, some preliminaries about the projected method are reviewed, and the new preconditioner for preconditioned GAOR method is introduced. Convergence analysis is given in Sect. 3. The convergence rates of the proposed preconditioned GAOR method are compared with the convergence rates of the preconditioned GAOR method in [13] and the convergence rates of the original GAOR method for LCP with an Mmatrix in Sect. 4, which shows that the proposed preconditioned GAOR converges faster than the preconditioned GAOR method in [13] and the original GAOR method. Numerical examples are given to verify the theoretical results in Sect. 5. Finally, conclusions are drawn in Sect. 6.
Preliminaries
We give some of the notations, definitions, and lemmas which will be used in the sequel. For \(A=(a_{i,j})\), \(B=(b_{i,j})\in R^{n\times n}\), we write \(A\geq B\) if \(a_{i,j}\geq b_{i,j}\) holds for all \(i,j=1,2, \ldots ,n\). \(A\geq O\) is called nonnegative if \(a_{i,j}\geq 0\) for all \(i,j=1,2,\ldots ,n\), where O is an \(n\times n\) zero matrix. For the vectors \(a, b\in R^{n\times 1}\), \(a\geq b\) and \(a\geq 0\) can be defined in a similar manner. By \(A=(a_{ij})\) we define the absolute value of a given matrix \(A\in \mathbb{R}^{n\times n}\). We denote the \(n\times n\) diagonal matrix coinciding in its diagonal with matrix \(C\in \mathbb{R}^{n\times n}\) by \(\operatorname{diag}(C)\). For simplicity, we may assume that \(a_{ii}=1\) (\(i=1,2 \dots n \)).
Lemma 1
([17])
Let \(A = [a_{i j}] \in R^{n \times n}\) and \(a_{i j} \leq 0\) for \(i\neq j\). A is an Mmatrix if and only if there exists a positive vector y such that \(Ay > 0\).
There are equivalent conditions of an Mmatrix, a nonsingular Mmatrix is a monotone matrix, see [4].
Definition 1
([4])
For a matrix \(A\in \mathbb{R}^{n\times n}\), the representation \(A=MN\) is called a splitting of A if M is nonsingular. Then \(A=MN\) is called weak regular if \(M^{1}\geq 0\) and \(M^{1}N\geq 0\).
For the weak regular splittings of different monotone matrices, there is a comparison result as follows.
Lemma 2
([4])
Suppose that \(A_{1}=M_{1}N_{1}\) and \(A_{2}=M _{2}N_{2}\) are weak regular splittings of the monotone matrices \(A_{1}\) and \(A_{2}\), respectively, such that \(M_{1}^{1}\leq M_{2} ^{1}\). If \(M_{1}^{1}N_{1}\) has a positive Perron vector, then
The following lemma is taken from [11].
Lemma 3
([11])
Let A be an Mmatrix and x be a solution of LCP (1). If \(f_{i}>0\), then \(x_{i}>0\) and therefore \(\sum^{n}_{j=1} a_{ij}x_{j}f_{i}=0\). Moreover, if \(f\leq 0\), then \(x=0\) is the solution of LCP (1).
For the study of the projected methods, the following definition is needed.
Definition 2
([15])
Given any vector \(x\in \mathbb{R}^{n}\), \(x_{+}\) denotes the vector with components
From Definition 2, the following properties hold for any \(x, y\in \mathbb{R}^{n}\):

(i)
\((x+y)_{+}\leq x_{+}+y_{+}\);

(ii)
\(x_{+}y_{+}\leq (xy)_{+}\);

(iii)
\(x=x_{+}+(x)_{+}\);

(iv)
\(x\leq y\Rightarrow x_{+}\leq y_{+}\).
Using Definition 2, LCP (1) is analogous to [1]
where \(D=\operatorname{diag}(A)\) is a nonsingular matrix. From (2) and the splitting of A as \(A=DLU\), here D, −L, −U are the diagonal, the strictly lower, and the strictly upper triangular parts of A, respectively. The GAOR method for solving (1) can be defined as (see [8, 12])
where \(\Omega =\operatorname{diag}(\omega_{1}, \omega_{2},\ldots ,\omega_{n})\) is a positive diagonal matrix and α is a positive real number. Let \(J=D^{1}(L+U)\), \(G=I\alpha \Omega D^{1}L\), and \(F=ID^{1}( \Omega A\alpha \Omega L)\), the following result concerns the convergence of the GAOR method for solving LCP (1).
Lemma 4
([12])
Suppose that A is a positive diagonal Hmatrix. Then, for any initial vector \(x^{0}\in R^{n}\), the iterative sequence \(\{x^{k} \}\) generated by the GAOR method (3) converges to the unique solution \(x^{*}\) of \(LCP\) (1), and
where \(0 < \omega_{i} < 2/[1 + \rho (J)]\) and \(0 \leq \alpha \leq 1 \).
For accelerating the convergence rate of the GAOR method, a preconditioner, based on the Hadjidimos preconditioner [7], is proposed in [13]
where \(\gamma_{i}\geq 0\) (\(i=2,\ldots ,n\)) and \(\beta_{i}\) (\(i=2,\ldots ,n\)) are constants. It has been showed that the preconditioned matrix \(\widetilde{A}=\widetilde{P}A\) is also an Mmatrix when A is an Mmatrix [13], hence the equivalent linear complementarity problem of LCP (1)
has the unique solution [3]. The preconditioned GAOR method for solving LCP (1) is defined [13] as
based on the splitting \(\widetilde{A}=\widetilde{D}\widetilde{L} \widetilde{U}\), please refer to [13] for more details.
Note that the preconditioning effect of P̃ is not observed on the first row of matrix A. In order to provide the preconditioning effect on all the rows of A, in this paper, we propose the following preconditioner:
where \(\gamma_{i}\geq 0\) (\(i=1,\ldots ,n\)) and \(\beta_{i}\) (\(i=1,\ldots ,n\)) are constants.
Let the preconditioned matrix \(\widehat{A}=\widehat{P}A\) be split as \(\widehat{A}=\widehat{D}\widehat{L}\widehat{U}\), where D̂, L̂, and Û are the diagonal, lower triangular, and upper triangular matrices, then the preconditioner GAOR method for solving LCP (1) is defined as
Convergence analysis
In this section, we will consider the convergence of the preconditioned GAOR method (6) for solving LCP (1). In what follows, we make the assumptions:
 (H0):

\(f_{1}>0\) and \(f_{n}>0\);
 (H1):

\(0\leq \gamma_{i}\leq 1\) for \(i=1,2,\ldots , n\);
 (H2):

\(\gamma_{1} a_{1n}+a_{1n}\leq \beta_{1}\leq \gamma_{1} a_{1n}\);
 (H3):

\(\gamma_{i} a_{i1}+a_{i1}\leq \beta_{i}\leq \gamma_{i} a_{i1} \) for \(i=2,3,\ldots ,n\).
Theorem 1
Let \(\widehat{A}=\widehat{P}A=[\hat{a}_{ij}]\) and \(\hat{f}= \widehat{P}f=[\hat{f}_{i}]\). If (H0)–(H3) are satisfied, then LCP (1) is equivalent to the linear complementarity problem
Proof
After some calculations, the elements of Â and f̂ can be expressed, respectively, as follows:
and
Let x be a solution of LCP (1). Since \(f_{1}>0\) and \(f_{n}>0\), from Lemma 3 we have \(x_{1}>0\), \(\sum^{n}_{j=1}a _{1j}x_{j}f_{1}=0\), and \(x_{n}>0\), \(\sum^{n}_{j=1}a_{nj}x_{j}f_{n}=0\). Therefore, on the one hand, if \(i=1\), then
On the other hand, if \(i\neq 1\), then
Thus, x is a solution of LCP (7).
Conversely, suppose that x is the solution to LCP (7), assumptions (H0), (H3) and (9) imply that \(\beta_{n} +\gamma_{n} a_{n1} \leq 0 \), so \((\beta_{n} +\gamma_{n} a _{n1})f_{1} \geq 0\), then \(f_{n}(\beta_{n} +\gamma_{n} a_{n1})f_{1} \geq 0\), we get that \(\hat{f}_{n}>0\). Similarly, from assumptions (H0), (H2) and (9), we can obtain \(\hat{f}_{1}>0\). It follows from Lemma 3 that \(x_{1}>0\), \(\sum^{n}_{j=1}\hat{a}_{1j}x_{j} \hat{f}_{1}=0\), and \(x_{n}>0\), \(\sum^{n}_{j=1}\hat{a}_{nj}x_{j} \hat{f}_{n}=0\). This together with (10) and (11) gives \(\sum^{n}_{j=1}a_{ij}x_{j}f_{i}=0\). Thus, for \(i=1\), we have
While for \(i\neq 1\), we can deduce that
Hence, x is the solution of LCP (1). □
Lemma 5
If A is an Mmatrix, (H1)–(H3) hold, then \(\widehat{A}= \widehat{P}A\) is an Mmatrix.
Proof
If A is an Mmatrix, then \(a_{ij}<0\) for \(i\neq j\) and \(a_{1i}a_{i1}<1\), which leads to \(a_{1i}>1/a_{i1}\). Otherwise, from (8) and assumption (H2), \(\gamma_{1} a_{1n}+a _{1n}\leq \beta_{1}\leq \gamma_{1} a_{1n}\), we have that \(\beta_{1} + \gamma_{1} a_{1n}\leq 0\) and \(\beta_{1} +\gamma_{1} a_{1n}\geq a_{1n}> 1/a_{n1}\). Then \(\hat{a}_{11}= a_{11}(\beta_{1} +\gamma_{1} a_{1n})a _{n1} > 11/a_{n1}\ast a_{n1}=0 \).
Similarly, we can obtain
From Lemma 1 there exists a positive vector \(y > 0\) such that \(Ay > 0\). Note that \(\widehat{P}>0\), thus \(\widehat{A} y = \widehat{P}Ay > 0\), and from Lemma 1, Â is an Mmatrix. □
Theorem 2
Let A be a diagonally dominant Mmatrix. If (H0)–(H3) hold, then for \(0\leq \omega_{i}\leq 2/[1+\rho (\hat{J})]\) (\(i=1,\ldots ,n\)) and \(0\leq \alpha \leq 1\), the iterative sequence of the preconditioned GAOR method (6) converges to the unique solution \(x^{\ast }\) of LCP (1), here \(\hat{J}=\widehat{D}^{1}(\widehat{L}+\widehat{U})\).
Proof
From Lemma 5, we know that Â is a diagonally dominant Hmatrix. Thus from Theorem 1 and Lemma 4, the preconditioned GAOR method (6) converges to the unique solution of LCP (1). □
Comparison results
In this section, we will compare the convergence rate of the preconditioned GAOR method (6) with that of the GAOR method (3) and that of the preconditioned GAOR method (4) [13]. For simplicity, we may assume that \(a_{ii}=1\), \(i=1,\ldots ,n\). For this case, \(G=I\alpha \Omega L\), \(F=I(\Omega A\alpha \Omega L)\), and D̃, L̃, and Ũ can be expressed as \(\widetilde{D}=IS_{D}\), \(\widetilde{L}=LS+S_{L}\), \(\widetilde{U}=U+S_{U}\) with
Moreover, the D̂, L̂, and Û can be expressed as \(\widehat{D}=IS_{D}R_{D}\), \(\widehat{L}=LS+S_{L}\), \(\widehat{U}=U+S_{U}R+R_{U}\) with
Lemma 6
If A is an Mmatrix with diagonal elements 1, (H1)–(H3) hold, then \(0\leq \widehat{D}\leq \widetilde{D}\).
Proof
Note that
and
Since A is an Mmatrix with diagonal elements 1, we have \(0< a_{ij}a_{ji}<1\) (\(i\neq j\)) [7]. Then assumptions (H1)–(H3) imply that
and
Hence, \(0\leq \widehat{D}\leq \widetilde{D}\). □
From Lemma 6 and the fact that \(\widehat{L}=\widetilde{L}\), we have
Moreover, it is easy to check that the inequality
holds [14]. Let \(\widetilde{G}=I\alpha \Omega \widetilde{D} ^{1}\widetilde{L}\), \(\widetilde{F}=I\widetilde{D}^{1}(\Omega \widetilde{A}\alpha \Omega \widetilde{L})\), \(\widehat{G}=I\alpha \Omega \widehat{D}^{1}\widehat{L}\), and \(\widehat{F}=I \widehat{D}^{1}(\Omega \widehat{A}\alpha \Omega \widehat{L})\). If (H1) and (H3) hold, then [14]
The following theorem gives a comparison result between \(\rho ( \widetilde{G}^{1}\widetilde{F})\) with \(\rho (\widehat{G}^{1} \widehat{F})\).
Theorem 3
If A is an irreducible nonsingular Mmatrix, and (H0)–(H3) hold, then
Proof
It follows from Theorem 2 and (14) that \(\rho (\widehat{G}^{1}\widehat{F})<1\) and \(\rho (\widetilde{G}^{1} \widetilde{F})<1\), so we only need to show
In terms of (12), we have
that is, \(\widehat{G}\leq \widetilde{G}\). Note that Ĝ and G̃ are Mmatrices, hence
As \(\widetilde{F}=I\widetilde{D}^{1}(\Omega \widetilde{A}\alpha \Omega \widetilde{L})\) and \(\widehat{F}=I\widehat{D}^{1}(\Omega \widehat{A}\alpha \Omega \widehat{L})\) are nonnegative matrices, thus, this together with (15) yields \(\widetilde{G}\widetilde{F}\) and \(\widehat{G}\widehat{F}\) are the regular splitting of different monotone matrices \(\widetilde{G}\widetilde{F}\) and \(\widehat{G} \widehat{F}\), respectively. Moreover, for a nonnegative matrix \(\widetilde{G}^{1}\widetilde{F}\), according to Perron–Frobenius theorem (see [4]), there is a positive Perron vector z such that
Hence, it follows from Lemma 2 that
The proof is completed. □
Remark 1
From Theorem 3 and (14), we can see that under assumptions (H0)–(H3), the inequalities
hold. This confirms that the proposed preconditioner P̂ in (5) is more efficient than the preconditioner P̃ [13] for accelerating the convergence rate of GAOR method for solving LCP (1) with an Mmatrix.
Numerical example
In this section, two examples are given for verifying the theoretical result.
Example 1
Linear complementarity problem with coefficient matrix
Tables 1 and 2 list \(\rho (G^{1}F)\), \(\rho (\widetilde{G}^{1} \widetilde{F})\), and \(\rho (\widehat{G}^{1}\widehat{F})\) for different α and Ω.
Example 2
Let the coefficient matrix A of LCP (1) be given by
where \(Q=(q_{ij})\in R^{p\times p}\), \(R=(r_{ij})\in R^{q\times q}\), \(L=(l_{ij})\in R^{q\times p}\), and \(U=(u_{ij})\in R^{p\times q}\) with
It is obvious that A is an irreducible Mmatrix.
Table 3 and Table 4 list \(\rho (G^{1}F)\), \(\rho (\widetilde{G}^{1}\widetilde{F})\), and \(\rho (\hat{G}^{1} \hat{F})\) with different α and Ω for Example 2, where in P̃, let \((\gamma_{1},\gamma_{2},\ldots ,\gamma_{n})^{T}=(0, \frac{1}{3},\ldots ,\frac{1}{3})^{T}\), \((\beta_{1},\beta_{1},\ldots , \beta_{n})^{T}=(0,0.003,\ldots ,0.003)^{T}\) and in P̂, let \(( \gamma_{1},\gamma_{2},\ldots ,\gamma_{n})^{T}=(\frac{1}{3}, \frac{1}{3},\ldots ,\frac{1}{3})^{T}\), \((\beta_{1},\beta_{1},\ldots , \beta_{n})^{T}=(0.003,0.003,\ldots , [4]0.003)^{T}\).
From Tables 1, 2, 3, and 4 we can see that, for different choices of α and Ω, the inequalities
hold, which verifies the theoretical result in Theorem 3.
Conclusions
In this paper, we present a new preconditioner P̂, which provides the preconditioning effect on all the rows of A, to accelerate the convergence rate of the GAOR method to solve LCP (1) with an Mmatrix A, and consider the preconditioned GAOR method (6). We prove that the original LCP (1) is equivalent to LCP (7), and show that the preconditioned GAOR method (6) is convergent for solving LCP (1). Then a comparison theorem on the preconditioned GAOR method (6) is obtained, which shows that the preconditioned GAOR method (6) improves the convergence rate of the preconditioned GAOR method in [13] for solving LCP (1). Together with the comparison result in [12], we know that the preconditioned GAOR method (6) improves considerably the convergence rate of the original GAOR method for solving LCP (1).
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Acknowledgements
The authors would like to thank the editors and reviewers for their valuable comments, which greatly improved the readability of this paper.
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This paper is supported by the China Postdoctoral Science Foundation (No.2017M613244).
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Miao, SX., Zhang, D. On the preconditioned GAOR method for a linear complementarity problem with an Mmatrix. J Inequal Appl 2018, 195 (2018). https://doi.org/10.1186/s1366001817895
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DOI: https://doi.org/10.1186/s1366001817895
MSC
 65F10
 65F15
Keywords
 Linear complementarity problem
 Preconditioner
 Preconditioned GAOR method
 Mmatrix