Iterative methods for nonlinear quasi complementarity problems
The author studies the generalized nonlinear quasi complementarity problem of finding \(u\in K(u)\) such that \(Tu\in K^*(u)\) and \((u- m(u),Tu)=0\), where m is a point-to-point mapping. T is a nonlinear continuous mapping from a real Hilbert space H into itself, \(K^*(u)\) is the polar cone of the convex cone K(u) in H, and (.,.) denotes the inner product. He gives an approximate solution to this problem by the following iterative scheme: (i) Given \(u_ 0\in H\). (ii) \(u_{n+1}=m(u_ n)+P_ k[u_ n-\rho (Tu_ n)-m(u_ n)]\), \(n=0,1,2,3,...\), where \(P_ k\) is the projection of H into K, and \(\tau >0\) is required to satisfy the following conditions: \[ | \rho - \frac{\alpha}{\beta^ 2}| < \frac{\sqrt{\alpha^ 2-4\beta^ 2(\gamma -\gamma^ 2)}}{\beta^ 2},\quad \gamma < 1/2 \text{ and } \alpha \geq 2\beta \sqrt{\gamma -\gamma^ 2}. \]
generalized nonlinear quasi complementarity problem
scientific article; zbMATH DE number 4012342