A multiplicative Schwarz iteration scheme for solving the linear complementarity problem with an \(H\)-matrix (Q999791)

The aim of this paper is to develop a multiplicative Schwarz iteration scheme to solve a linear complementarity problem consisting in finding \(x \in \mathbb{R}^n\) such that \(x \geqslant \varphi\), \(Ax - F \geqslant 0\), and \((x-\varphi)^T(Ax-F) = 0\), where \(A \in \mathbb{R}^{n \times n}\) is a given \(H_+\)-matrix and \(\varphi, F \in \mathbb{R}^n\) are given vectors. This numerical method is an extension of the multiplicative Schwarz iteration scheme for solving the associated linear equation \(Ax-F = 0\) proposed by \textit{R. Bru, F. Pedroche} and \textit{D. B. Szyld} [Linear Algebra Appl. 393, 91--105 (2004; Zbl 1066.65036)]. The authors establish the (monotone) convergence of the sequence generated by the multiplicative Schwarz iteration scheme under appropriate assumptions and analyze the convergence rate for different overlapping sizes. The efficiency of the proposed algorithm is illustrated by numerical experiments.