A generalized Newton method of high-order convergence for solving the large-scale linear complementarity problem
- Yajun Xie^{1, 2} and
- Changfeng Ma^{1, 2}Email author
https://doi.org/10.1186/s13660-015-0937-4
© Xie and Ma 2015
Received: 8 October 2015
Accepted: 8 December 2015
Published: 23 December 2015
Abstract
In this paper, by extending the classical Newton method, we present the generalized Newton method (GNM) with high-order convergence for solving a class of large-scale linear complementarity problems, which is based on an additional parameter and a modulus-based nonlinear function. Theoretically, the performance of high-order convergence is analyzed in detail. Some numerical experiments further demonstrate the efficiency of the proposed new method.
Keywords
MSC
1 Introduction
The linear complementarity problem frequently arises in various scientific and engineering applications, such as Nash equilibrium point of a bimatrix game, the contract problem, and the free boundary problem for journal bearings; see [1–3].
In [3], Lemke proposed first a solution for linear complementarity problem. Along these ideas, Scarf has given the approximation of fixed-points of a continuous mapping [4]. The relationship between the linear complementarity problem and the fixed-points problem is well described by Eaves et al. [5, 6].
Many efficient methods were developed to solve linear complementarity problem. Especially, when the system matrix A is large and sparse. For instance, we have the projected successive overrelaxation iteration [7] and the general fixed-point iterations [8]. On the matrix splitting iterations approaches, Bai et al. have derived some fruitful research results [9–12], especially, in [9], Bai proposed the modulus-based matrix splitting iteration scheme which is a powerful method for solving the linear complementarity problem. Matrix multisplitting iteration aspects, also, in many works were considered by Bai et al. to solve the linear complementarity problem in [13–17]. A variety of accelerated modulus-based matrix splitting iteration versions were also established; see [18, 19]. Furthermore, the modulus-based synchronous multisplitting iteration methods for large sparse linear complementarity problems are introduced in [20]. On the basis of [20], Ljiljana et al. avoided the assumption of the parameter constraint and improved the convergence area [21]. Recently, in [22], by the vector divisions and the secant method, Foutayeni et al. investigated an efficient hybrid method for solving the linear complementarity problem.
As is well known the semismooth (or smooth) Newton method is very efficient for some nonsmooth (or smooth) equations, which arise from the complementarity problem, the nonlinear programming problem, the variational inequality problem, the discretization problem of partial differential equations, the maximal monotone operator problem, etc.; see [23–25] for a detailed discussion. These methods are competitive since they converge rapidly for any sufficiently right initial guess. In order to ensure that the global convergence of semismooth Newton methods, some merit functions, such as squared norm merit functions, are often exploited; see [26] and the references therein. However, just as the statement in [27], a globalization of the semismooth Newton method for nonsmooth equations is very hard because the corresponding merit function is nondifferentiable in most cases.
In view of this, by introducing a smooth equation and some reasonable equivalent reformulations, we investigate a generalized Newton iteration method with high-order convergence rate for solving a class of large-scale linear complementarity problem, which make full use of the superiority of the second-order convergence rate of the classical Newton method. In this article, we suppose that the matrix A of the linear complementarity problem (1.1) is a P-matrix, i.e., the determinants of all principal submatrices are positive. Under this assumption, as is well known, the linear complementarity problem (1.1) has a unique solution for every q.
For simplicity of the presentation, we use the following notations throughout the paper: Let \(\mathbb{N}_{k}=\{1,2,\ldots,k\}\) denote the set of first k positive integers. For \(x\in\mathbb{R}^{n}\), \(\|x\|\) stands for the Euclidean norm. Given two real \(n\times m\) matrices \(A=(a_{ij})\) and \(B=(b_{ij})\), we write \(A\geq B\) (or \(A>B\)) if \(a_{ij}\geq b_{ij}\) (or \(a_{ij}>b_{ij}\)) hold for all \(i\in\mathbb {N}_{n}\) and \(j\in\mathbb{N}_{m}\). \(|A|\) and \(\rho(A)\) denote the absolute value and spectral radius of the matrix \(A\in \mathbb{R}^{n\times m}\), respectively. For a differential function \(F(x)\), \(F'(x)\) is referred to as the Jacobi matrix of the function \(F(x)\). For an invertible matrix A, \(A^{-1}\) denotes the inverse matrix of A. The matrix \(\operatorname{Diag}\{ a_{1}, a_{2}, \ldots, a_{n}\}\) denotes the diagonal matrix, where \(a_{i}\) (\(i\in \mathbb{N}_{n}\)) are the elements of the principal diagonal.
The outline of the paper is organized as follows. In Section 2, we first consider a generalized Newton method (GNM) with high-order convergence rate for solving a class of the linear complementarity problems (1.1). In Section 3, we analyze the performance and rate of convergence of the GNIM in detail. Some numerical experiments are given to illustrate that the GNM is efficient in Section 4. At last, we end the paper with some conclusions in Section 5.
2 The generalized Newton method
We first introduce some useful results which are vital for the equivalent reformulation as regards the \(LCP(q,A)\). Furthermore, these conclusions contribute significantly to the analysis of the convergence rate of the generalized Newton iteration method which will be presented in the following.
Lemma 2.1
Proof
Lemma 2.2
Lemma 2.3
Proof
Now, we show the generalized Newton iteration method for the nonlinear smooth system (2.7). A detailed description follows.
Algorithm 2.1
(The generalized Newton method)
Step 1 Input the initial guess \(z^{0}=(\varepsilon_{0}, (x^{0})^{T})^{T}\), give the matrix A and vector q and any small positive numbers \(\sigma_{1}, \sigma_{2}\in(0,1)\), preset a positive integer \(m\geq2\). Set \(k:=0\).
Step 2 Compute \(\Psi(z^{k})\), the Jacobi matrix \(\Psi'(z^{k})\), and its inverse matrix \(A_{k}:=(\Psi'(z^{k}))^{-1}\).
Step 3 Set \(z^{k,1}=z^{k}\), \(j:=1\).
Step 5 Set \(j:=j+1\), \(z^{k,j}=z^{k,j+1}\), \(\Psi(z^{k,j})=\Psi (z^{k,j+1})\), \(y:=A_{k}b\). If \(j=m\), return to Step 6, otherwise go to Step 4.
Step 6 If \(\|y\|<\sigma_{1}\) or \(\|\Psi(z^{k,m})\|<\sigma_{2}\), let \(z^{k,m}=z^{*}\), otherwise \(k:=k+1\), return to Step 3.
Remark 2.1
From Lemma 2.2, we know that the iterative solution \(z^{*}\) generated by Algorithm 2.1 is also the solution \(u^{*}\) of \(LCP(q,A)\) (1.1).
Remark 2.2
The update of parameter \(\varepsilon_{k}\) can be chosen with \(\varepsilon _{k}=\varepsilon_{k-1}^{m}\). Since the positive integer m is selected at least greater than or equal to 2 in Algorithm 2.1, the positive sequence \(\{\varepsilon_{k}\}_{0}^{\infty}\) declines monotonically and tends to zero.
Remark 2.3
Once we set \(m=1\), then the GNIM reduces to the classical Newton iterative method.
3 The analysis of convergence
Definition 3.1
([28])
Let \(F: D\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\), \(x^{*}\in D\) is the solution of system \(F(x)=0\). There is a region \(S\subset D\) for the point \(x^{*}\), for any initial approximation \(x^{0}\in S\), if the iteration sequence \(\{x^{k}, k=0,1,\ldots\}\) is always well defined and converges to \(x^{*}\), we call it the attractive point of the iteration sequence.
The classical Newton iteration features a convergence rate of at least order two. We have the following results; for more details see [28] and the references therein.
Lemma 3.1
([28])
Lemma 3.2
([28])
Lemma 3.3
[28]
We are now in a position to derive the main convergence result of the generalized Newton iteration method.
Theorem 3.1
Proof
Now, we state that the iterative scheme (2.12) has at least a convergence rate of order \(m+1\). The result will be shown by mathematical induction.
4 Numerical experiments
In fact, our approach can be considered as the version of speeding up on the basis of the smoothing Newton method. Thereby, high precision is the advantage of the GNIM. Once we set \(m=1\), the GNIM will reduce to the classical Newton method. But we know that the larger m may lead to the consumption of more CPU time since there is an increased number of inner iterations. We usually choose \(m=2\) in concrete tests, which also can ensure the rapid convergence rate. To confirm this judgment, we can observe the following examples.
Example 4.1
([27])
Example 4.2
We consider the linear complementarity problem (1.1) with \(A=(a_{ij})\), \(i, j\in\mathbb{N}_{n}\), where \(a_{ij}=\frac{i\delta _{ij}}{n}\), δ is the Kronecker delta (\(\delta_{ii}=1\), \(\delta_{ij}=0\), when \(i\neq j\)), and \(q=(q_{i})\), \(i\in \mathbb{N}_{n}\) such that \(q_{i}=-1\).
The GNIM numerical results for Example 4.1 with initial \(\pmb{z_{(1)}^{0}}\)
The performance of numerical results | The solution pairs \(\boldsymbol {(u^{*},w^{*})}\) | ||
---|---|---|---|
n = 3 | It | 3 | u∗ = (0.3571,0.4286,0.3571) |
CPU | 0.0068 | ||
RES | 0 | w∗ = (0,0,0) | |
n = 5 | It | 3 | u∗ = (0.3654,0.4615,0.4808,0.4615,0.3654) |
CPU | 0.0073 | ||
RES | 0 | w∗ = 1.0e − 015∗(0.4441,0,0,0,0) | |
n = 8 | It | 3 | \(\begin{array}{lcl}u*&=&(0.3660,0.4641,0.4902,0.4967,\\ &&{}0.4967,0.4902,0.4641,0.3660) \end{array}\) |
CPU | 0.0074 | ||
RES | 0 | w∗ = (0,0,0,0,0,0,0,0) | |
n = 10 | It | 3 | \(\begin{array}{lcl}u*&=&(0.3660,0.4641,0.4904,0.4974,0.4991,\\ &&{}0.4991,0.4974,0.4904,0.4641,0.3660) \end{array}\) |
CPU | 0.0075 | ||
RES | 0 | \(\begin{array}{lcl} w*&=&1.0e\!-\!015*(-0.1110,0.4441,-0.4441,\\ &&{}0,0,0,0.2220,0,0,0) \end{array}\) |
The CBSN numerical results for Example 4.1 with initial \(\pmb{z_{(1)}^{0}}\)
The performance of numerical results | The solution pairs \(\boldsymbol {(u^{*},w^{*})}\) | ||
---|---|---|---|
n = 3 | It | 4 | u∗ = (0.3571,0.4286,0.3571) |
CPU | 0.0104 | ||
RES | 0 | w∗ = 1.0e − 011∗(0.8880,−0.5869,0.8880) | |
n = 5 | It | 4 | u∗ = (0.3654,0.4615,0.4808,0.4615,0.3654) |
CPU | 0.0106 | ||
RES | 6.1063e − 016 | w∗ = 1.0e − 015∗(0,0,0,0,−0.1110) | |
n = 8 | It | 4 | \(\begin{array}{lcl} u*&=&(0.3660,0.4641,0.4902,0.4967,0.4967,\\ &&{}0.4902,0.4641,0.3660) \end{array}\) |
CPU | 0.0113 | ||
RES | 6.2042e − 016 | \(\begin{array}{lcl} w*&=&1.0e\!-\!011*(0.4839,-0.1609,-0.0001,0,0,\\ &&{}-0.0001,-0.1608,0.4839) \end{array}\) | |
n = 10 | It | 3 | \(\begin{array}{lcl} u*&=&(0.3660,0.4641,0.4904,0.4974, 0.4991,\\ &&{}0.4991,0.4974,0.4904,0.4641,0.3660) \end{array}\) |
CPU | 0.0115 | ||
RES | 6.2232e − 016 | \(\begin{array}{lcl} w*&=&1.0e\!-\!011*(0.4835,-0.1607,-0.0001,0,0,\\ &&{}0,0,-0.0001,-0.1607,0.4835) \end{array}\) |
Numerical results for Example 4.1
Initials | \(\boldsymbol {z^{0}_{(1)}}\) | \(\boldsymbol {z^{0}_{(1)}}\) | \(\boldsymbol {z^{0}_{(2)}}\) | \(\boldsymbol {z^{0}_{(2)}}\) | |
---|---|---|---|---|---|
Methods | GNIM | CBSN | GNIM | CBSN | |
n = 50 | It | 2 | 3 | 3 | 4 |
CPU | 0.0084 | 0.0250 | 0.0119 | 0.0447 | |
RES | 5.7787e − 016 | 1.7554e − 016 | 5.3787e − 016 | 1.8619e − 016 | |
n = 100 | It | 2 | 3 | 3 | 4 |
CPU | 0.0135 | 0.0230 | 0.0142 | 0.0364 | |
RES | 6.1815e − 016 | 2.1678e − 016 | 5.6610e − 016 | 2.7616e − 016 | |
n = 300 | It | 2 | 3 | 3 | 4 |
CPU | 0.0376 | 0.0547 | 0.0562 | 0.0905 | |
RES | 6.2315e − 016 | 2.2648e − 016 | 5.7610e − 016 | 2.9616e − 016 |
Numerical results for Example 4.1
Initials | \(\boldsymbol {z^{0}_{(1)}}\) | \(\boldsymbol {z^{0}_{(1)}}\) | \(\boldsymbol {z^{0}_{(2)}}\) | \(\boldsymbol {z^{0}_{(2)}}\) | |
---|---|---|---|---|---|
Methods | GNIM | FBSN | GNIM | FBSN | |
n = 500 | It | 2 | 11 | 3 | 11 |
CPU | 0.0886 | 0.3229 | 0.1366 | 0.3831 | |
RES | 6.1815e − 016 | 6.6613e − 016 | 5.6610e − 016 | 1.1047e − 015 | |
n = 1000 | It | 2 | 11 | 3 | 11 |
CPU | 0.5114 | 1.0758 | 0.7728 | 1.3069 | |
RES | 6.2113e − 016 | 6.6723e − 016 | 5.6720e − 016 | 1.3027e − 015 | |
n = 3000 | It | 2 | 11 | 3 | 11 |
CPU | 9.1418 | 26.6155 | 11.1484 | 11.5429 | |
RES | 6.2812e − 016 | 6.6921e − 016 | 5.5610e − 016 | 1.2123e − 015 | |
n = 5000 | It | 2 | 11 | 3 | 11 |
CPU | 42.9507 | 71.3675 | 66.0831 | 79.3341 | |
RES | 6.4615e − 016 | 6.6990e − 016 | 5.7810e − 016 | 1.3056e − 015 |
The GNIM numerical results for Example 4.2 with initial \(\pmb{z_{(1)}^{0}}\)
The performance of numerical results | The solution pairs \(\boldsymbol {(u^{*},w^{*})}\) | ||
---|---|---|---|
n = 3 | It | 2 | u∗ = (3,1.5,1) |
CPU | 0.0067 | ||
RES | 0 | w∗ = (0,0,0) | |
n = 5 | It | 2 | u∗ = (5,2.5,1.6667,1.25,1) |
CPU | 0.0070 | ||
RES | 0 | w∗ = 1.0e − 015∗(0.2220,0,0,0,0) | |
n = 8 | It | 2 | u∗ = (8,4,2.6667,2,1.6,1.3333,1.1429,1) |
CPU | 0.0072 | ||
RES | 0 | w∗ = (0,0,0,0,0,0,0,0) | |
n = 10 | It | 2 | \(\begin{array}{lcl} u*&=&(10,5,3.3333,2.5,2,1.6667,1.4286,\\ &&{}1.25,1.1111,1) \end{array}\) |
CPU | 0.0075 | ||
RES | 0 | \(\begin{array}{lcl}w*&=&1.0e\!-\!015*(0,0.2220,-0.1110,0,0,\\ &&{}0,0.2220,0,0,0) \end{array}\) |
The CBSN numerical results for Example 4.2 with initial \(\pmb{z_{(1)}^{0}}\)
The performance of numerical results | The solution pairs \(\boldsymbol {(u^{*},w^{*})}\) | ||
---|---|---|---|
n = 3 | It | 4 | u∗ = (3,1.5,1) |
CPU | 0.0155 | ||
RES | 0 | w∗ = (0,0,0) | |
n = 5 | It | 4 | u∗ = (5,2.5,1.6667,1.25,1) |
CPU | 0.0161 | ||
RES | 0 | w∗ = 1.0e − 015∗(0.4441,0,0,0,0) | |
n = 8 | It | 4 | u∗ = (8,4,2.6667,2,1.6,1.3333,1.1429,1) |
CPU | 0.0163 | ||
RES | 2.2204e − 016 | \(\begin{array}{lcl} w*&=&1.0e\!-\!011*(-0.1110,0.2220,0,0,\\ &&{}0.2220,0,0,0) \end{array}\) | |
n = 10 | It | 4 | \(\begin{array}{lcl} u*&=&(10,5,3.3333,2.5,2,1.6667,\\ &&{}1.4286,1.25,1.1111,1) \end{array}\) |
CPU | 0.0147 | ||
RES | 8.8818e − 016 | \(\begin{array}{lcl} w*&=&1.0e\!-\!014*(-0.1221,0.0444,-0.0111,\\ &&{}0.0222,0,-0.0111,0.0222,0,0,0) \end{array}\) |
Numerical results for Example 4.2
Initials | \(\boldsymbol {z^{0}_{(1)}}\) | \(\boldsymbol {z^{0}_{(1)}}\) | \(\boldsymbol {z^{0}_{(2)}}\) | \(\boldsymbol {z^{0}_{(2)}}\) | |
---|---|---|---|---|---|
Methods | GNIM | FBSN | GNIM | FBSN | |
n = 500 | It | 2 | 11 | 3 | 11 |
CPU | 0.0957 | 0.3025 | 0.1421 | 0.3308 | |
RES | 2.3182e − 015 | 1.7073e + 001 | 2.1006e − 015 | 1.8073e + 001 | |
n = 1000 | It | 2 | 11 | 3 | 11 |
CPU | 0.5016 | 0.9461 | 0.6327 | 1.1754 | |
RES | 6.6438e − 014 | 2.4129e + 001 | 6.6351e − 014 | 2.6249e + 001 |
To sum up, by all numerical results, we illustrate the effectiveness of the GNIM.
5 Conclusion
In this paper, we establish the generalized Newton method (GNM) for solving the large-scale linear complementarity problem. The GNM features the high-order convergence rate, at least convergence order \(m+1\), which has been verified from theoretical analysis section in detail. The new strategy will increase the efficiency remarkably. In fact, it can be regarded as an accelerated process for the classical Newton approach. Experimental tests provide the comparison of the numerical performance for some existing efficient methods, which testify to the viability of the proposed approach.
Declarations
Acknowledgements
The Project Supported by National Natural Science Foundation of China (Grant Nos. 11071041, 11201074), Fujian Natural Science Foundation (Grant Nos. 2015J01578, 2013J01006) and The Outstanding Young Training Plan of Fujian Province universities (Grant No. 15kxtz13, 2015).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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