On convergence of the modulus-based matrix splitting iteration method for horizontal linear complementarity problems of H+-matrices

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Abstract

Horizontal linear complementarity problem has wide applications, such as in mechanical and electrical engineering, structural mechanics, piecewise linear system, telecommunication systems and so on. In this paper, we focus on the convergence conditions of the modulus-based matrix splitting iteration method proposed recently for solving horizontal linear complementarity problems. By the proposed theorems, the assumptions on the matrix splitting and the system matrices are weakened, and the convergence domain is enlarged. Numerical examples are presented to show the improvement.

Introduction

For two given matrices A,BRn×n and qRn, the horizontal linear complementarity problem, abbreviated as HLCP, is to find two real vectors z,wRn such thatz0,w0,Az=Bw+q,andwTz=0,where for two m × n real matrices P=(pij) and Q=(qij) 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 H+-matrices is proposed. Comparing to the results in [15], the contributions of this work are given below:

  • weaken the assumptions on the system matrices and corresponding matrix splittings;

  • 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 A=(aij), let |A|=(|aij|). 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 |aii|>ji|aij| 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 A10. Denote A=(aij) the comparison matrix of A, where aij=|aij| if i=j and aij=|aij| 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 H+-matrix (e.g., see [33]). Call A=MN an M-splitting if M is a nonsingular M-matrix and N is nonnegative. A=MN is called an H-splitting if M|N| is a nonsingular M-matrix; and an H-compatible splitting if A=M|N| (e.g., see [22]). Below are some known results:

  • 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]).

  • 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]).

  • if A=MN is a nonsingular M-splitting of A, then ρ(M1N)<1 holds (see [32]).

  • for an H-matrix A, the fact that A=MN is an H-compatible splitting is a sufficient but not necessary condition of that A=MN 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

[15]

Let A=MANB and B=MBNB be two splittings of ARn×n and BRn×n, 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 x=12(γzΩ1w) satisfies:(MA+MBΩ)x=(NA+NBΩ)x+(BΩA)|x|+γq;

(2) If x is given by (2), then (z, w) is a solution of (1), wherez=1γ(x+|x|),w=1γΩ(|x|x).

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 InRn×n and ImRm×m be the identity matrices, A=A^+μIn,B=B^+νIn and q=Az*Bw*, where n=m2, A^=blktridiag(Im,S,Im)Rn×n,B^=ImSRn×n,S=tridiag(1,4,1)Rm×m and μ, ν are real parameters.

Example 3.8. Let InRn×n and ImRm×m be the identity matrices, A=A^+μIn,B=B^+νIn and q=Az*Bw*, where n=m2, A^=blktridiag(1.5Im,S,0.5Im)Rn×n,B^=

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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