Convergence results of a matrix splitting algorithm for solving weakly nonlinear complementarity problems
- Mei-Ju Luo^{1}Email author,
- Ya-Yi Wang^{1} and
- Hong-Ling Liu^{1}
https://doi.org/10.1186/s13660-016-1139-4
© Luo et al. 2016
Received: 13 April 2016
Accepted: 28 July 2016
Published: 22 August 2016
Abstract
In this paper, we consider a class of weakly nonlinear complementarity problems (WNCP) with large sparse matrix. We present an accelerated modulus-based matrix splitting algorithm by reformulating the WNCP as implicit fixed point equations based on two splittings of the system matrixes. We show that, if the system matrix is a P-matrix, then under some mild conditions the sequence generated by the algorithm is convergent to the solution of WNCP.
Keywords
weakly nonlinear complementarity problems modulus-based matrix splitting algorithm convergence analysisMSC
90C33 65F10 65L101 Introduction
As is well known, the classical nonlinear complementarity problems (NCP) are very important and fundamental topics in optimization theory and they have been developed into a well-established and fruitful principle. See [1–3] for details as regards the basic theories, effective algorithms and important applications of NCP. Problem (1) is a special case of NCP, but it extends from linear complementarity problems. When \(\Psi(z)=q\) is a constant vector, problem (1) reduces to a linear complementarity problem. Recently, lots of researchers [4–8] have paid close attention to feasible and efficient methods for solving linear complementarity problems. Especially, by reformulating linear complementarity problems as an implicit fixed point equation, Van Bokhoven [9] proposed a modulus iteration method, which is defined as the solution of system of linear equations at each iteration. In 2010, Bai [10] presented a modulus-based matrix splitting iteration method and showed the convergence when the system matrix is an \(H_{+}\)-matrix. Consequently, Zhang [11] proposed two-step modulus-based matrix splitting iteration methods and considered the convergence theory when the system matrix is an \(H_{+}\)-matrix. Based on the above work, for solving problem (1), in [12] Sun and Zeng proposed a modified semismooth Newton method with A being an M-matrix and \(\Psi(z)\) being a continuously differentiable monotone diagonal function on \(R^{n}\).
In this paper, for satisfying the requirements of the application, more details can be found in [13–16], we present an accelerated modulus-based matrix splitting algorithm for dealing with WNCP. The organization of this paper is as follows: Some necessary notations and definitions are introduced in Section 2. In Section 3, we establish a class of accelerated modulus-based matrix splitting iteration algorithms. In Section 4, the convergence conditions are considered.
2 Preliminaries
Some necessary notations, definitions and lemmas used in the sequel discussions are introduced in this section. For \(B\in R^{n\times n}\), we write \(B^{-1}\), \(B^{T}\), \(\rho(B)\) to denote the inverse, the transpose, the spectral radius of the matrix B, respectively. For \(x\in R^{n}\), we write \(\Vert x\Vert \), \(\vert x\vert \) to denote the norm of the vector x, \(\vert x\vert =(\vert x_{1}\vert ,\ldots, \vert x_{n}\vert )\), respectively. \(\Vert A\Vert \) denotes any norm of matrix A. Especially, we use \(\Vert \cdot \Vert _{2}\) to denote a spectral norm. \(\lambda\in\lambda(A)\) denotes the eigenvalue of matrix A where \(\lambda(A)\) is the set of all eigenvalues of matrix A.
Definition 1
[17]
If for any \(x:=(x_{1},x_{2},\ldots,x_{n})\neq0\), there exists an index k, such that \(x_{k}(Ax)_{k}=x_{k}(a_{k1}x_{1}+\cdots+a_{kn}x_{n})>0\), we call that matrix A is a P-matrix.
Definition 2
Lemma 3
[18]
Let \(A=(a_{ij})\in R^{n\times n}\) be a P-matrix, for any nonnegative diagonal matrix Ω, the matrix \(A+\Omega\) is nonsingular.
3 Algorithm
Theorem 4
- (i)If z is a solution of problem (1), then \(x=\frac {1}{2} (\Gamma^{-1}z-\Omega^{-1}(Az+\Psi(z)) )\) satisfies the implicit fixed point equation$$ (M_{1}\Gamma+\Omega_{1})x =(N_{1}\Gamma-\Omega_{2})x+(\Omega-M_{2} \Gamma)\vert x\vert +N_{2}\Gamma \vert x\vert -\Psi \bigl(\Gamma \bigl(\vert x\vert +x\bigr) \bigr). $$(2)
- (ii)
Proof
Algorithm 5
Step 1. Choose two splittings of the matrix \(A\in R^{n\times n}\) satisfying \(A=M_{1}-N_{1}=M_{2}-N_{2}\).
Step 2. Set \(k=0\). Give an initial vector \(x^{0}\in R^{n}\), compute \(z^{0}=\frac{1}{\gamma}(\vert x^{0}\vert +x^{0})\).
Step 4. If the sequence \(\{z^{k}\}_{0}^{+\infty}\) is convergent, stop. Otherwise, go to Step 3.
4 Convergence theorems
In this section, we will consider the conditions that ensure the convergence of \(\{z^{k}\}_{0}^{+\infty}\) obtained by Algorithm 5.
Theorem 6
Proof
Theorem 7
- (i)When \(0<\tau_{1}+\tau_{2}<\frac{\lambda _{\min}-L}{\lambda_{\max}}\),$$\omega=\sqrt{\lambda_{\max}\lambda_{\min}}. $$
- (ii)When \(\frac{\lambda_{\min}-L}{\lambda_{\max}} <\tau _{1}+\tau_{2}<\frac{\lambda_{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}\),$$\sqrt{\lambda_{\max}\lambda_{\min}}\leq\omega< \frac{[1-(\tau_{1}+\tau _{2})]\lambda_{\max}\lambda_{\min}-L\lambda_{\max}}{(\tau_{1}+\tau _{2})\lambda_{\max}+L-\lambda_{\min}}. $$
- (iii)When \(\tau_{1}+\tau_{2}=\frac{\lambda _{\min}-L}{\lambda_{\max}}\),$$\omega\geq\sqrt{\lambda_{\max}\lambda_{\min}}. $$
Proof
5 Results and discussion
This study focused on the weakly nonlinear complementarity problems with a large sparse matrix. We proposed an algorithm that is not only computationally more convenient to use but also faster than the modulus-based matrix splitting iteration methods and the convergence conditions are presented when the system matrix is a P-matrix.
Some scholars had already stressed the accelerated modulus-based matrix splitting iteration methods for linear complementarity problems and pointed out that the system matrix is either a positive definite matrix or an \(H_{+}\)-matrix. However, we suggest that the system matrix is a P-matrix, this is more adaptable but also a limitation. Notwithstanding its limitation, this study does suggest that WNCP can be solved faster.
6 Conclusions
In this paper, by reformulating the complementarity problem (1) as an implicit fixed point equation based on splittings of the system matrix A, we establish an accelerated modulus-based matrix splitting iteration algorithm and show the convergence analysis when the involved matrix of the WNCP is a P-matrix.
Declarations
Acknowledgements
This work was supported in part by NSFC Grant NO. 11501275 and Scientific Research Fund of Liaoning Provincial Education Department NO. L2015199.
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
References
- Lin, GH, Fukushima, M: New reformulations for stochastic nonlinear complementarity problems. Optim. Methods Softw. 21, 551-564 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Tseng, P: Growth behavior of a class of merit functions for the nonlinear complementarity problem. J. Optim. Theory Appl. 89, 17-37 (1996) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, C, Chen, XJ: Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty. J. Optim. Theory Appl. 137, 277-295 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Chen, XJ, Zhang, C, Fukushima, M: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51-80 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Fang, H, Chen, X, Fukushima, M: Stochastic \(R_{0}\) matrix linear complementarity problems. SIAM J. Optim. 18, 482-506 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Fukushima, M: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 137, 99-110 (1992) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, C, Chen, XJ: Smoothing projected gradient method and its application to stochastic linear complementarity problems. SIAM J. Optim. 20, 627-649 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Zhou, G, Caccetta, L: Feasible semismooth Newton method for a class of stochastic linear complementarity problems. J. Optim. Theory Appl. 139, 379-392 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Van Bokhoven, WM: A class of linear complementarity problems is solvable in polynomial time. Technical report, Department of Electrical Engineering, Technical University, Eindhoven (1980) Google Scholar
- Bai, Z: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 6, 917-933 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, L: Two-step modulus based matrix splitting iteration for linear complementarity problems. Numer. Algorithms 57, 83-99 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Sun, Z, Zeng, J: A monotone semismooth Newton type method for a class of complementarity problems. J. Comput. Appl. Math. 235, 1261-1274 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Chung, K: Equivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems. Academic Press, New York (1992) Google Scholar
- Fukushima, M: Merit functions for variational inequality and complementarity problems. In: Nonlinear Optimization and Applications, vol. 137, pp. 155-170 (1996) View ArticleGoogle Scholar
- Lin, GH, Chen, XJ, Fukushima, M: New restricted NCP function and their applications to stochastic NCP and stochastic MPEC. Optimization 56, 641-753 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Ling, C, Qi, L, Zhou, G: The \(SC^{1}\) property of an expected residual function arising from stochastic complementarity problems. Oper. Res. Lett. 36, 456-460 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Schafer, U: A linear complementarity problem with a P-matrix. SIAM Rev. 46, 189-201 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Berman, A, Plemmons, R: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979) MATHGoogle Scholar