A two-step modulus-based matrix splitting iteration method for solving nonlinear complementarity problems of $$H_+$$
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A two-step modulus-based matrix splitting iteration method for solving nonlinear complementarity problems of H+ -matrices Hua Zheng1
· Ling Liu1
Received: 29 January 2018 / Revised: 16 April 2018 / Accepted: 11 May 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract In this paper, we establish a two-step modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems with the system matrix being an H+ -matrix. The convergence analysis of the proposed method is given. Numerical examples show that the proposed method is efficient. Keywords Nonlinear complementarity problem · Two-step modulus-based method · H+ -matrix Mathematics Subject Classification 65F10 · 90C33
1 Introduction In this paper, we consider the nonlinear complementarity problem (NCP( f )) for finding a vector z ∈ Rn so that f (z) = Az + ϕ(z) ≥ 0, z ≥ 0,
and z T f (z) = 0,
(1)
where A = (ai j ) ∈ Rn×n , ϕ(z) is a nonlinear function and for two s × t matrices K = (ki j ) and T = (ti j ) the order K ≥ (>)T means ki j ≥ (>)ti j for any i and j. If ϕ(z) = q ∈ Rn , NCP (1) reduces to the linear complementarity problem (LCP(q, A)), which arises in the free boundary problems, the network equilibrium problems and the contact problems, etc. (e.g., see Murty 2008; Cottle et al. 1992 and the references therein). Recently, some solvers of LCP(q, A) based on the following modulus equation have been given:
Communicated by Jinyun Yuan.
B
Hua Zheng [email protected] Ling Liu [email protected]
1
School of Mathematics and Statistics, Shaoguan University, Shaoguan, People’s Republic of China
123
H. Zheng, L. Liu
(Ω2 + AΩ1 )x = (Ω2 − AΩ1 )|x| − q, where Ωi , i = 1, 2, are positive diagonal parameter matrices. In particular, Bai proposed a modulus-based matrix splitting iteration method for solving LCP(q, A) and presented convergence analysis for the proposed methods; see Bai (2010). Moreover, it gives rise to the modulus-based matrix splitting relaxation methods such as Jacobi, Gauss–Seidel, SOR, and AOR, which are practical and effective in actual implementations. The two-step modulusbased matrix splitting and the two-step modulus-based synchronous multisplitting iteration methods of H -matrices were established in Zhang (2011) and Zhang (2015), respectively. For more discussions and further generalizations of the modulus-based matrix splitting iteration methods, we refer to recent studies Li (2013), Bai and Zhang (2013a, b), Zheng and Yin (2013), Zhang and Ren (2013), Cvetkovi´c, and Kosti´c (2014), Zhang (2014), Cvetkovi´c et al. (2014), Zheng and Li (2015), Liu et al. (2016), Li and Zheng (2016) and Zheng et al. (2017) and references therein. If ϕ(z) is a general function, then NCP (1) belongs to nonlinear complementarity problems, see Ferris and Pang (1997) and Harker and Pang (1990), which have wide applications. The modulus-based iteration methods are also used to solve NCP (1). In Xia and Li (2015) and Huang and Ma (2016), the modulus-based matrix splitting iteratio
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