Abstract
In this paper, we present an efficient method for finding a numerical solution for nonlinear complementarity problems (NCPs). We first reformulate an NCP as an equivalent system of fixed-point equations and then present a modulus-based matrix splitting iteration method. We prove the convergence of the proposed method with theorems with the relevant conditions. Our preliminary numerical results show that the method is feasible and effective.
1. Introduction
In this paper, we consider the following nonlinear complementarity problem with a nonlinear source term, namely, finding a such thatwhere the function has the form of , is a given large, sparse, and real matrix, is a given vector, is a given diagonal differentiable mapping, that is, the th component of is a function of the th variable only
Clearly, if is a linear function, problem (1) reduces to a linear complementarity problem (LCP).
Nonlinear complementarity problem (NCP) has many applications in nonlinear compressed sensing, signal processing, communications, DNA microarrays, and so on [1–7]. For example, the nonlinear compressed sensing theory has been widely applied in asteroseimology for significant detection. The polynomial structure has been employed in many applications cases, such as quadratic measurements in sparse signal recovery and nonlinear compressed sensing with polynomial measurements. In this regard, under some constraint qualifications, the original problem turns out to finding the sparsest solutions to a special NCP. NCP can also be derived from discrete simulations of the Bratu obstacle problem and free boundary problems with nonlinear source terms. This problem has received much attention during the past two decades and has been studied extensively with applicable numerical methods to obtain an approximated solution. There are different kinds of numerical methods that have been developed, including the classical linearized projected relaxation method [8], multilevel method [9], domain decomposition method [10, 11], penalty method [12–14], and semismooth Newton method [15]. Most of those methods need to solve the linear complementarity subproblems; see [16–19] for several typical iteration methods.
It is well known that the matrix splitting method is a very important and effective method for computing systems of linear equations. Many authors have extended this kind of method for LCP. Bai [20] presented a unified framework for the construction of modulus-based matrix splitting iteration methods. Since then, many scholars have developed various kinds of modulus-based matrix splitting iteration methods; see, for example, [21–23] for LCP and [24, 25] for NCP. Recently, Wu and Li [26] proposed a class of new modulus-based matrix splitting methods for LCP. This kind of method is different from previously published works. The new modulus-based matrix splitting iteration methods provide a new general framework for solving the LCP. Numerical experiments show that the new modulus-based matrix splitting iteration method for LCP is feasible and competitive when compared with other well-known modulus-based matrix splitting iteration methods. Motivated by the ongoing work in this field, in this paper, we extend the new modulus-based matrix splitting method for a kind of NCP and establish its convergence through theorems.
The remainder of this paper is structured as follows. In Section 2, we present some relevant preliminary information and the modulus-based matrix splitting iteration method. We establish the convergence theorems of the method under suitable conditions in Section 3. In Section 4, we present numerical results to show the efficiency of the proposed method.
2. Modulus-Based Matrix Splitting Iteration Method
We briefly introduce some definitions and lemmas that will be used in the paper. Most of them can be found in [20, 25, 27, 28].
Given two matrices , , we call if holds for all , . If for all , , is called a nonnegative matrix, denoted by . denotes the nonnegative matrix with entries .
Let be a real matrix. Its comparison matrix , where
The matrix is called a matrix if its off-diagonal entries are all nonpositive; is called an matrix if is a matrix and ; is called an matrix if its comparison matrix is an matrix; is called an matrix if it is an matrix with positive diagonal entries; if is an matrix, is a positive diagonal matrix, and then implies that is an matrix. We use to represent the spectral radius of the square matrix .
Given , if is a nonsingular matrix, then is called a splitting of the matrix ; for the splitting , if is an matrix, the splitting is called a splitting.
Lemma 1 (see [27]). Let with . If there exists with such that , then .
Lemma 2 (see [28]). Let be an matrix and be a matrix with . Then, is an matrix.
Lemma 3 (see [28]). Let be an matrix, be the diagonal part of the matrix , and . Then, matrices and are nonsingular, and .
Lemma 4 (see [26]). Let be an splitting of the matrix , where is an matrix. Then, , where and .
To obtain a new equivalence system of fixed-point equations of the NCP, we need the following lemma:
Lemma 5 (see [26]). Let . Then,if and only if
This result carries immediately over to the vectors in .
Based on Lemma 5, we have the following theorem:
Theorem 1. Let be a positive diagonal matrix. Then, NCP (1.1) is equal to
Proof. Obviously, NCP (1.1) is equal towhere is a positive diagonal matrix. Here, we take and for Lemma 5 and getThe proof is completed.
We take as a matrix splitting of matrix ; then, from (6), we haveThis new equivalent expression (9) is different from the one in the previous work in [24, 25]. These methods have similarities but they do not belong to each other. Based on (9), we can present the following modulus-based matrix splitting iteration method for solving (1).
Method 1 (modulus-based matrix splitting iteration method). Let be a splitting of the matrix and be a nonsingular matrix, where is a positive diagonal matrix. Given an initial vector , for until the iteration sequence is convergent, compute byMethod 1 provides a general framework of the modulus-based matrix splitting iteration method for solving the NCP. Based on the matrix splitting technique, some new modulus-based relaxation methods are obtained. For example, we express matrix aswhere and and are the strictly lower and upper triangular matrices of , respectively. Then,(i)When , , , from Method 1, we have which is called a modulus iteration (MI) method.(ii)When , , from Method 1, we have which is called the modulus-based AOR (MAOR) iteration method. Clearly, the MAOR method becomes the new modulus-based successive overrelaxation, Gauss–Seidel, and Jacobi method when is equal to , (1, 1), and (1, 0), respectively.
3. Convergence Theory
In this section, we discuss the convergence of Method 1 under suitable conditions.
Theorem 2. Let be an positive diagonal matrix, be a splitting of a matrix , with being a nonsingular matrix. Suppose that there exist positive constants such that , , for all . Let be a diagonal matrix, and the diagonal elements are , . Let
If , then the iteration sequence generated by Method 1 converges to the solution of NCP (1.1).
Proof. Let be the solution for the NCP (1.1). Then, from (6), we obtainBased on (10) and (15), noting that is nonsingular, we haveThis indicates thatIt is clear that when , the iteration sequence generated by Method 1 converges to the solution of NCP (1.1). This completes the proof.
Similar to the proof of Theorem 2, using for (3.2), we have the following theorem.
Theorem 3. Let be an positive diagonal matrix, let be a splitting of a matrix , with being a nonsingular matrix. Suppose that there exist positive constants such that , , for all . Let be a diagonal matrix, and the diagonal elements are , . Let
If , then the iteration sequence generated by Method 1 converges to the solution of the NCP (1.1).
Theorem 4. Let be an splitting of the matrix , where is an matrix. Let the diagonal matrix , where . Suppose there exist positive constants such that , , for all . Let be a diagonal matrix, and the diagonal elements are , . Let . If is an matrix, then the iteration sequence generated by Method 1 converges to the solution of the NCP (1.1).
Proof. Since is an splitting andMatrix is an matrix from Lemma Method 2. Noting that , by Lemma 3, we can see is an matrix andMoreover, we havewhere the last inequality holds by Lemma 4. Since is an matrix, there exists a positive vector such thatHence,We have by Lemma 1, and the iteration sequence generated by Method 1 converges to the solution of NCP (1.1). This completes the proof.
4. Numerical Experiments
In this section, we test the effectiveness of the proposed method on two problems. We implemented our methods in MATLAB and ran the code on a PC with a 3.4 GHz CPU and 8GB RAM. We tested the proposed method in Section 2 and compared their performance with Method 4.1 proposed in [24].
Method 2. Let be a splitting of the matrix , be an nonnegative diagonal matrix, and be a positive number. Given , for , compute by solving the linear system:and setThe iteration terminates when the sequence is convergent.
In all the experiments, we chose the same and in Method 2 as in [24], that is, and .
We considered the following two problems:
Problem 1. Consider (1.1), let , where is an matrix, and are matrices with , and a diagonal mapping and .
Problem 2. Consider (1.1), let , where is an matrix, and are matrices with , and and a diagonal mapping , and .
For the above problems, the initial points of all the methods were set to . Define Res and the stopping criterion for all the methods are set to Res . We can obtain many matrix splitting iteration methods from Methods 1 and 2 using different splittings of . In our experiments, we consider two splittings of matrix :(i)Let and denote the method corresponding to Method 1 as and the method corresponding to Method 2 as , respectively.(ii)Let and , where are the diagonal, the strictly lower triangle, and the strictly upper triangle of the matrix , respectively, and is a relaxation parameter. Denote the methods corresponding to Method 1 as and Method 2 as , respectively.We first test the effect of the choice of matrix in Method 1 on Problem 1. We let and fix the dimension of the problem . The results are listed in Tables 1 and 2. The column of the table has the following meaning.
Table 1 is the numerical result for , and Table 2 is the numerical result for with fixed . As we can see from the tables, has a great influence on the behaviour of the proposed method. In the following experiments, we fix . We compared our Method 1 with Method 2 proposed in [24]. The results are listed in Tables 3–6. “—” means the method failed to find the solution. From the tables, we can see that our method outperforms the method proposed in [24] regardless of the number of iterations or the CPU elapsed time. In these two examples, seems to be competitive with with the best parameters. In addition, as we can see from Tables 4 and 6, methods and may be independent of the dimensions.
5. Conclusions
In this paper, we have presented an efficient class of modulus-based matrix splitting methods for NCPs, which are based on an implicit system of fixed-point equations. A number of sufficient conditions to guarantee the convergence of the novel iteration method are presented. The proposed method is easy to implement, and in each iteration, there is no need to solve the nonlinear equations or linear complementarity subproblems; hence, the calculation cost is small. The comparisons of the numerical results show the computational efficiency of the proposed method. There are still many studies that need to be done. For example, we can investigate the two-step modulus-based matrix splitting method, that is, in each iteration, the method implements a forward sweep followed by a backward sweep. We can expect the two-step method to achieve a higher computing efficiency. Moreover, we can apply this new method to other complementarity problems, such as the implicit complementarity problems, and discuss its convergence (Table 5).
Data Availability
All the datasets used in this paper are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.