On convergence of the modulus-based matrix splitting iteration method for horizontal linear complementarity problems of -matrices
Introduction
For two given matrices and the horizontal linear complementarity problem, abbreviated as HLCP, is to find two real vectors such thatwhere for two m × n real matrices and the order P ≥ ( > )Q means pij ≥ ( > )qij for any i and j.
HLCP is an important generalization of linear complementarity problem (LCP) and arises in the fields of mechanical and electrical engineering, structural mechanics, piecewise linear system, telecommunication systems, inventory theory, optimization and statistics; see [1], [2], [3], [4], [5], [6] for details.
Several methods had been devised for solving HLCP, such as interior-point methods [7], [8], reduction to LCP [9], [10], homotopy approaches [11], neural networks [12], verification methods [13] and projected splitting methods [14]. Recently, in [15] the modulus-based matrix splitting (MMS) iteration method was established and numerical experiments showed that this kind of methods is a powerful tool for solving HLCP. In recent years, the classes of the MMS methods had already been successfully used for solving other complementarity problems, such as LCP [16], [17], [18], [19], [20], [21], [22], [23], nonlinear complementarity problem [24], [25], implicit complementarity problem [26], [27], quasi-complementarity problem [28] and second-order cone linear complementarity problem [29]. On the other hand, the modulus method is applied to the field of inverse problems; see [30], [31].
In this paper, new convergence analysis of the MMS method for solving HLCP of -matrices is proposed. Comparing to the results in [15], the contributions of this work are given below:
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weaken the assumptions on the system matrices and corresponding matrix splittings;
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obtain a larger convergence domain of the parameter positive diagonal matrix.
The improvements of the new theorems are verified by the numerical examples.
Next, we present necessary notations, concepts and basic results first. The notations e and “⊗” stand for an n × 1 vector of ones and the Kronecker product, respectively. For an n × n real matrix let . By tridiag(a, b, c) we denote a tridiagonal matrix with constants a, b, c as its subdiagonal, main diagonal and superdiagonal entries, respectively, while by blktridiag(A, B, C) we denote a block tridiagonal matrix with matrices A, B, C as its corresponding block entries. If for all i, A is called a strictly diagonally dominant (s.d.d.) matrix. If aij ≤ 0 for any i ≠ j, we call A a Z-matrix. A is called a nonsingular M-matrix if A is a nonsingular Z-matrix and . Denote the comparison matrix of A, where if and if i ≠ j. A is called an H-matrix if ⟨A⟩ is a nonsingular M-matrix. These concepts can be found in [32]. An H-matrix with positive diagonal entries is called an -matrix (e.g., see [33]). Call an M-splitting if M is a nonsingular M-matrix and N is nonnegative. is called an H-splitting if is a nonsingular M-matrix; and an H-compatible splitting if (e.g., see [22]). Below are some known results:
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if A is an s.d.d. Z-matrix with aii > 0 for all 1 ≤ i ≤ n, then A is a nonsingular M-matrix (see [32]).
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if A is a nonsingular M-matrix, then there exists a positive diagonal matrix D such that AD is an s.d.d. matrix (see [32]).
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if is a nonsingular M-splitting of A, then holds (see [32]).
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for an H-matrix A, the fact that is an H-compatible splitting is a sufficient but not necessary condition of that is an H-splitting (see [22], [23]).
Note that the existence and uniqueness of the solution of HLCP had been discussed in [4]. We always assume that the HLCP considered in this paper has a unique solution.
The organization of this paper is as follows. In Section 2 the new convergence theorems are given. Numerical results are reported in Section 3 to verify the new results.
Section snippets
Improved convergence theorems
By [15], we know that (1) is equivalent to a modulus equation with a lemma given below: Lemma 2.1 Let and be two splittings of and respectively. Let Ω be a n × n positive diagonal matrix and γ be a positive constant. Then, the following statements hold true: (1) If (z, w) is a solution of (1), then satisfies: (2) If x is given by (2), then (z, w) is a solution of (1), where[15]
Based on (2), the MMS method for
Numerical examples
In this section, numerical experiments are provided to show the advantages of the proposed theorems.
Consider the following two examples from [15].
Example 3.7. Let and be the identity matrices, and where and μ, ν are real parameters.
Example 3.8. Let and be the identity matrices, and where
Acknowledgment
The authors would like to thank the reviewers for their helpful suggestions. This work is supported by the National Natural Science Foundation of China with No. 11601340, University of Macau with No. MYRG2018-00047-FST, The Science and Technology Development Fund, Macau SAR (File no. 0005/2019/A), the Major Projects of Guangdong Education Department for Foundation Research and Applied Research with No. 2018KZDXM065 and the Young Innovative Talents Project from Guangdong Provincial Department of
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