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Relaxed Modulus-Based Matrix Splitting Methods for the Linear Complementarity Problem †

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Relaxed Modulus-Based Matrix Splitting Methods for the Linear Complementarity Problem † S S symmetry Article Relaxed Modulus-Based Matrix Splitting Methods for the Linear Complementarity Problem † Shiliang Wu 1, *, Cuixia Li 1 and Praveen Agarwal 2,3 1 School of Mathematics, Yunnan Normal University, Kunming 650500, China; [email protected] 2 Department of Mathrematics, Anand International College of Engineering, Jaipur 303012, India; [email protected] 3 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, United Arab Emirates * Correspondence: [email protected] † This research was supported by National Natural Science Foundation of China (No.11961082). Abstract: In this paper, we obtain a new equivalent fixed-point form of the linear complementarity problem by introducing a relaxed matrix and establish a class of relaxed modulus-based matrix split- ting iteration methods for solving the linear complementarity problem. Some sufficient conditions for guaranteeing the convergence of relaxed modulus-based matrix splitting iteration methods are presented. Numerical examples are offered to show the efficacy of the proposed methods. Keywords: linear complementarity problem; matrix splitting; iteration method; convergence MSC: 90C33; 65F10; 65F50; 65G40 1. Introduction Citation: Wu, S.; Li, C.; Agarwal, P. Relaxed Modulus-Based Matrix In this paper, we focus on the iterative solution of the linear complementarity problem, Splitting Methods for the Linear abbreviated as ‘LCP(q, A)’, whose form is Complementarity Problem. Symmetry T 2021, 13, 503. https://doi.org/ w = Az + q ≥ 0, z ≥ 0 and z w = 0, (1) 10.3390/sym13030503 × where A 2 Rn n and q 2 Rn are given, and z 2 Rn is unknown, and for two s × t matrices Academic Editor: Jan Awrejcewicz G = (gij) and H = (hij) the order G ≥ (>)H means gij ≥ (>)hij for any i and j. As is known, as a very useful tool in many fields, such as the free boundary problem, the Received: 12 February 2021 contact problem, option pricing problem, nonnegative constrained least squares problems, Accepted: 11 March 2021 see [1–5], LCP(q, A) is most striking in the articles, see [6–11]. Published: 19 March 2021 Designing iteration methods to fast and economically computing the numerical so- lution of the LCP(q, A) is one of the hotspot nowadays, which were widely discussed in 1 Publisher’s Note: MDPI stays neutral the articles, see [1,2,8,9,12,13] for more details. Recently, by using w = g W(jxj − x) and with regard to jurisdictional claims in 1 z = g (jxj + x) with g > 0 for the LCP(q, A), Bai in [14] expressed the LCP(q, A) as the published maps and institutional affil- fixed-point form iations. (W + M)x = Nx + (W − A)jxj − gq, (2) where A = M − N is a splitting of matrix A and W denotes a positive diagonal matrix, and then first desgined a class of modulus-based matrix splitting (MMS) iteration methods. Since the MMS method has the advantages of simple form and fast convergence rate, it was Copyright: © 2021 by the authors. q A Licensee MDPI, Basel, Switzerland. regarded as a powerful method of solving the LCP( , ), and raises concerns. For several This article is an open access article variants and applications of the MMS method, one can see [2,15–32]. distributed under the terms and In this paper, based on the MMS method, we will create a type of new iteration conditions of the Creative Commons methods to solve the LCP(q, A). Our strategy is to introduce a relaxed matrix for both Attribution (CC BY) license (https:// sides of Equation (2) and obtain a new equivalent fixed-point form of the LCP(q, A). Based creativecommons.org/licenses/by/ on this new equivalent form, we can establish a class of relaxed modulus-based matrix 4.0/). splitting (RMMS) iteration methods for solving the LCP(q, A). This class of new iteration Symmetry 2021, 13, 503. https://doi.org/10.3390/sym13030503 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 503 2 of 14 methods not alone inherits the virtues of the modulus-based methods. A more important one is that we can choose the relaxed matrix to enhance the computational efficiency of the classical MMS method in [14]. To guarantee the convergence of the RMMS iteration method, some sufficient conditions will be given under suitable conditions. Numerical examples are provided to verify that the RMMS iteration method are feasible and overmatch the classical MMS iteration method in terms of the computational efficiency. The layout of this paper is as follows. In Section2, for the sake of discussion in the rest of this paper, we provide some necessary definitions, lemmas and notations. In Section3, we present a class of relaxed modulus-based matrix splitting (RMMS) iteration methods to solve the LCP(q, A). The convergence conditions of the RMMS iteration method are presented in Section4. Numerical experiments with regard to the proposed methods compared to the classical MMS iteration method are reported in Section5. In Section6, we summarize this work. Finally, some simple discussions are given in Section7. 2. Preliminaries Some necessary definitions, lemmas and notations, which are used in the sequel discussions, are primitively introduced in this section. n×n Let A = (aij) 2 R . Then it is named as a Z-matrix if aij ≤ 0 (i 6= j); a nonsingular −1 M-matrix if A is a Z-matrix and A ≥ 0; its comparison matrix is hAi = (haiij) with haiij being ( jaijj for i = j, haiij = i, j = 1, 2, . , n. −jaijj for i 6= j, n×n Further, a matrix A = (aij) 2 R is named as an H-matrix if hAi = (haiij) is an M-matrix; an H+-matrix if A is an H-matrix with diag(A) > 0; a P-matrix if all of its principal minors are positive [33,34]. In addition, we let jAj = (jaijj). × Let A = M − N be a splitting of matrix A 2 Rn n. Then it is named as an M-splitting if M is a nonsingular M-matrix and N ≥ 0; an H-splitting if hMi − jNj is a nonsingular M-matrix. As is known, if A = M − N is an M-splitting and A is a nonsingular M-matrix, then r(M−1N) < 1, where r(·) indicates the spectral radius (the maximum value of the absolute value of the eigenvalues) of the matrix, see [33,34]. Finally, k · k2 denotes the Euclidean norm on Rn. n×n n Lemma 1 ([19]). Let A = (aij) 2 R with aij ≥ 0. If there exists u 2 R with u > 0 such that Au < u, then r(A) < 1. × Lemma 2 ([35]). Let A 2 Rn n be an H-matrix, D be the diagonal part of the matrix A, and A = D − B. Then matrices A and jDj are nonsingular, jA−1j ≤ hAi−1 and r(jDj−1jBj) < 1. Lemma 3 ([34]). Let A be an M-matrix and B be a Z-matrix with A ≤ B. Then B is an M-matrix. In addition, there exists a famous result for the existence and uniqueness of the LCP(q, A), that is to say, the LCP(q, A) has a unique solution if and only if a matrix A is a P-matrix, see [1]. Obviously, when A is an H+-matrix, the LCP(q, A) has a unique solution as well. 3. Relaxed Modulus-Based Matrix Splitting Method In this section, we introduce a class of relaxed modulus-based matrix splitting (RMMS) iteration methods for solving the LCP(q, A). For this purpose, by introducing an identical × equation Rx − Rx = 0 for Equation (2), where R 2 Rn n is a given relaxed matrix, then we obtain (W + M)x = Nx + (W − A)jxj + R(x − x) − gq, (3) or, (W + M − R)x = (N − R)x + (W − A)jxj − gq. (4) Symmetry 2021, 13, 503 3 of 14 Based on Equations (3) and (4), we can establish the following iteration method, which is named as a class of relaxed modulus-based matrix splitting (RMMS) iteration methods for the LCP(q, A). × Method 1. Let A = M − N be a splitting of the matrix A 2 Rn n. Let g > 0 and matrix × W + M − R be nonsingular, where W is a positive diagonal matrix and R 2 Rn n is a given relaxed ( ) ( + ) matrix. Given an initial vector x 0 2 Rn, compute z k 1 2 Rn 1 z(k+1) = (jx(k+1)j + x(k+1)), k = 0, 1, 2, . , g where x(k+1) can be obtained by solving the linear system (W + M)x(k+1) = Nx(k) + (W − A)jx(k)j + R(x(k+1) − x(k)) − gq (5) or (W + M − R)x(k+1) = (N − R)x(k) + (W − A)jx(k)j − gq. (6) Clearly, when R = 0, method1 reduces to the well-known MMS iteration method in [14]. Similarly, by introducing an identical equation, Wu and Li in [22] presented two- sweep modulus-based matrix splitting iteration methods for the LCP(q, A). The goal of introducing the relaxed matrix R in (5) or (6) is that the computational efficiency of the RMMS iteration method may be better than the classical MMS iteration method in [14]. In fact, method1 is a general framework of RMMS iteration methods for solving the LCP(q, A). This implies that we can construct some concrete forms of RMMS iteration methods by the specific splitting matrix of matrix A and the iteration parameters. If we take 1 1 M = (D − bL) and N = [(1 − a)D + (a − b)L + aU], a a where D, L and U, respectively, are the diagonal, the strictly lower-triangular and the strictly upper-triangular parts of the matrix A, then this leads to the relaxed modulus- based AOR (RMAOR) iteration method (W + D − bL − R)x(k+1) = [(1 − a)D + (a − b)L + aU − R]x(k) + (W − aA)jx(k)j − gaq, (7) (k+1) 1 (k+1) (k+1) with z = g (jx j + x ).
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