A class of matrix splitting-based fixed-point iteration method for the vertical nonlinear complementarity problem
Wang Yapeng, Mu Xuewen
TL;DR
This work addresses solving the VNCP, which requires finding $x$ such that $u(x)=Ax+\phi(x)\ge0$, $v(x)=Bx+\psi(x)\ge0$, and $u(x)^T v(x)=0$, using a matrix-splitting-based fixed-point framework. By introducing an auxiliary variable $y=| (A-\Omega B)x+(\phi(x)-\Omega\psi(x)) |$ and decomposing $A+\Omega B=M-N$, the authors derive an iterative scheme that updates $x^{k+1}=M^{-1}[Nx^k+y^k-\phi(x^k)-\Omega\psi(x^k)]$ and $y^{k+1}=(1-\tau)y^k+\tau |(A-\Omega B)x^{k+1}+\phi(x^{k+1})-\Omega\psi(x^{k+1})|$, with Jacobi, Gauss-Seidel, and SOR variants arising from different splittings. They establish two convergence results: (i) under $\alpha(\beta+\gamma)<1$ and a $\tau$-range, convergence to a unique VNCP solution; (ii) a second norm-based result with a stricter $\tau$-bound using a weighted error norm, yielding an explicit iteration-count estimate. Numerical experiments demonstrate that the FPI approach outperforms the prior modulus-based splitting method in iteration count and CPU time across several VNCP instances, underscoring the method's practical efficiency. The results provide a theoretically grounded, flexible, and efficient tool for VNCPs with potential broad applicability in scientific computing and engineering.
Abstract
In this paper, we propose a class of matrix splitting-based fixed-point iteration (FPI) methods for solving the vertical nonlinear complementarity problem (VNCP). Under appropriate conditions, we present two convergence results obtained using different techniques and estimate the number of iterations required for the FPI method. Additionally, through numerical experiments, we demonstrated that the FPI method surpasses other methods in computational efficiency.