Advances in Numerical Optimisation: Theory, Models, and Applications
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CuiXia Li, ShiLiang Wu, "Generalized SORLike Iteration Method for Linear Complementarity Problem", Mathematical Problems in Engineering, vol. 2020, Article ID 6314798, 6 pages, 2020. https://doi.org/10.1155/2020/6314798
Generalized SORLike Iteration Method for Linear Complementarity Problem
Abstract
In this paper, we present a generalized SORlike iteration method to solve the nonHermitian positive definite linear complementarity problem (LCP), which is obtained by reformulating equivalently the implicit fixedpoint equation of the LCP as a twobytwo block nonlinear equation. The convergence properties of the generalized SORlike iteration method are discussed under certain conditions. Numerical experiments show that the generalized SORlike method is efficient, compared with the SORlike method and the modulusbased SOR method.
1. Introduction
The linear complementarity problem is to find such thatwhere is a given matrix and is a given vector, which is abbreviated as the LCP(). Since the LCP() of form (1) often occurs in many actual problems of scientific computing and engineering applications, such as the linear and quadratic programming, the economies with institutional restrictions upon prices, the optimal stopping in Markov chain, and the free boundary problems, its numerical solution attracts considerable attention. For more detailed descriptions, one can refer to [1–6] and the references therein.
Recently, from the point of view of the system of the linear equations, some efficient numerical methods for solving the large and sparse LCP() are developed. Especially, based on the implicit fixedpoint equation of the LCP(), a class of modulus iteration method in [7] (see Section 9.2 in [3] as well) and its various versions have been presented in the literature. The goal of modulus iteration method is to take and such that the LCP() can be equivalently transformed into a system of fixedpoint equations:
In this way, based on fixedpoint equation (2), the modulus iteration method is described as follows.
Modulus Iteration Method. Given an initial vector , compute by solving the following linear system:
Then, setuntil the iteration sequence is convergent.
Making the simple substitution for in (3) results in the modified modulus iteration method, which was considered in [8]. Numerical results in [8] showed that the modified modulus method is feasible when the involved matrix is symmetric positive definite. In [9], combining the modulus method with the matrix splitting of the matrix , a class of modulusbased matrix splitting iteration methods is developed, which not only includes some presented iteration methods, such as the modified modulus method [8] and nonstationary extrapolated modulus algorithms [10] but also yields a series of iteration methods, such as modulusbased Jacobi, GaussSeidel, SOR, and AOR iteration methods. Further discussing the modulusbased matrix splitting iteration method and its various versions, one can see [11–17] for more details. In addition, for other forms of iteration methods, one can see [18–22].
In this paper, we focus on this situation where the involved matrix of the LCP() in (1) is nonHermitian positive definite. By reformulating equivalently the implicit fixedpoint equation of the LCP() as a twobytwo block nonlinear equation, based on the GSOR iteration method in [23], we extend the GSOR iteration method for the LCP() in (1) with its twobytwo block form. That is to say, we present a generalized SORlike iteration method to solve the LCP(). The convergence conditions of the generalized SORlike iteration method are discussed under suitable choices of the involved parameter. Numerical examples are reported to show that the generalized SORlike iteration method is feasible and effective in computing.
For our analysis, here we briefly explain some terminologies used in the next section. Let be the finite dimension Euclidean space, whose norm is denoted by . For , denotes a vector with components equal to depending on whether the corresponding component of is positive, zero, or negative. The diagonal matrix denotes a diagonal matrix corresponding to . Matrix is called a nonHermitian positive definite matrix if its Hermitian part, , is positive definite.
This paper is organized as follows. In Section 2, the generalized SORlike iteration method is established and its convergence properties are discussed. In Section 3, the generalized SORlike iteration method is used to solve the absolute value equation (AVE). Numerical experiments are reported in Section 4, and finally, some concluding remarks are given in Section 5.
2. Generalized SORLike Iteration Method
In this section, the generalized SORlike iteration method is established. To this end, we take and , where is a nonnegative diagonal matrix, and then the LCP() can be equivalently transformed into the following fixedpoint equations:
Let . From (5), we obtainthat is,where .
Letwhere
Then, the iteration scheme of the generalized SORlike iteration method iswhere and . Furthermore, the generalized SORlike method can be described as follows.
The Generalized SORLike Iteration Method. Let be a nonHermitian positive definite matrix, be a nonnegative diagonal matrix, and . Given initial vectors and , for until the iteration sequence is convergent, compute
When in (11), the generalized SORlike iteration method reduces to the SORlike method [24]. That is to say, the generalized SORlike iteration method is a generalization form of the SORlike method [24].
Let be the solution pair of the equation (7) and be generated by the iteration method (11). Let the iteration errors be
Then, we give the following main result with respect to generalized SORlike iteration method (11).
Theorem 1. Let be nonHermitian positive definite andIfthenwhereThis implies that the generalized SORlike method is convergent.
Proof. By the simple computations, from equations (7) and (11), we haveFrom (17), we can obtainFurthermore,Letand denote the spectral radius of matrix . When , the generalized SORlike method (11) converges. Let be an eigenvalue of the matrix . Then, satisfieswhich is equal toApplying Lemma 2.1 in [25] for equation (23), is equivalent toTherefore, if condition (14) holds, then . This completes the proof.
Clearly, when , then the generalized SORlike method reduces to the SORlike method. For the SORlike method, we have the following corollary.
Corollary 1. Let be nonHermitian positive definite andIfthenwhereThis implies that the SORlike method is convergent.
3. Generalized SORLike Method for AVE
On the basis of Proposition 2 in [26], the linear complementarity problem (LCP) and the absolute value equations (AVEs) are equivalent under certain conditions. Based on this, in this section, we will extend the generalized SORlike method for the following AVE:where , , and denotes all the components of the vector by absolute value.
Based on the results in Section 2, it is easy to obtain that the generalized SORlike method for AVE (29) can be established and described as follows.
The Generalized SORLike Iteration Method for the AVE. Let be nonsingular and . Given initial vectors and , for until the iteration sequence is convergent, compute
It is easy to see that we use instead of in Theorem 1, so the convergence condition of the generalized SORlike iteration method for AVE (29) is obtained.
Theorem 2. Let be nonsingular and . Ifthenwherethis implies that the generalized SORlike method is convergent.
Furthermore, for the SORlike method, we have the following corollary.
Corollary 2. Let be nonsingular and . DenoteIfthenwhereThis implies that the SORlike method is convergent.
Comparing Corollary 2 with Theorem 3.1 in [24], it is easy to see that the region of the parameter in Corollary 2 is the same as that in Theorem 3.1 in [24]. Both require in Corollary 2 and Theorem 3.1 in [24]. The difference between Corollary 2 and Theorem 3.1 in [24] is on and . The former is , and the latter issee Theorem 3.1 in [24]. Formally, the former is simpler than the latter.
4. Numerical Experiments
In this section, two examples are given to illustrate the feasibility and effectiveness of the generalized SORlike method in terms of iteration steps (denoted by “IT”) and computing time (denoted by “CPU”). Here, all initial vectors are chosen to beAll iterations are terminated once RES(), where “RES” is defined aswith being the kth approximate solution to the LCP and the minimum being taken componentwise in [9]. All the tests are performed in MATLAB 7.0.
To show the advantage of the generalized SORlike method, we compare the numerical results of the generalized SORlike method with the SORlike method [24] and the modulusbased SOR method [9].
In our computations, for the sake of convenience, we take in the generalized SORlike method, the SORlike method [24], and the modulusbased SOR method [9].
In the following tables, “GSOR” denotes the generalized SORlike method, “SOR” denotes the SORlike method [24], “MSOR” denotes the modulusbased SOR method [9], and “” denotes the CPU times larger than 500 seconds or the iteration numbers larger than 500 steps.
Example 1 (see [9]). Let LCP() in (1) withbe the unique solution of the LCP().
In Tables 1 and 2, for different problem sizes of , we list the iteration steps, the CPU times with the generalized SORlike method, the SORlike method, and the modulusbased SOR method. From the numerical results in Tables 1 and 2, we observe that the modulusbased SOR method fails to converge in 500 iterations. The generalized SORlike method and the SORlike method converge and quickly compute a satisfactory approximation to the solution of the LCP(). Furthermore, it is easy to see that the generalized SORlike method requires less iteration steps than the SORlike method. Moreover, the generalized SORlike method costs less CPU times than the SORlike method. Therefore, in terms of computing efficiency, the generalized SORlike method outperforms both the SORlike method and the modulusbased SOR method under certain conditions.


Example 2 (see [9]). Let LCP() in (1) withbe the unique solution of the LCP().
In Tables 3 and 4, for different problem sizes of , we list the iteration steps and the CPU times with respect to the generalized SORlike method, the SORlike method, and the modulusbased SOR method. These numerical results further confirm the observations obtained from Tables 1 and 2, i.e., the generalized SORlike method is superior to both the SORlike method and the modulusbased SOR method in terms of computing efficiency under certain conditions.


5. Conclusion
In this paper, we have presented a generalized SORlike iteration method for solving the nonHermitian positive definite linear complementarity problem (LCP) in (1), which is obtained by reformulating equivalently the implicit fixedpoint equation of the LCP as a twobytwo block nonlinear equation. Some convergence properties of the generalized SORlike iteration method are obtained. That is, the generalized SORlike iteration method can converge to the solution of the LCP in (1) under suitable choices of the involved parameter. Numerical experiments have been reported to confirm the efficiency of the proposed method.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (No. 11961082).
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Copyright © 2020 CuiXia Li and ShiLiang Wu. 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.