HYBRID METHOD FOR DESIGNING EXPLICIT HIERARCHICAL FIXED POINT APPROACH TO MONOTONE VARIATIONAL INEQUALITIES

Abstract

Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$. Assume that $F: C \to H$ is a $\kappa$-Lipschitzian and $\eta$-strongly monotone operator with constants $\kappa,\eta \gt 0$, $f: C \to H$ is $L$-Lipschitzian with constant $L \geq 0$ and $T,V: C \to C$ are nonexpansive mappings with ${\rm Fix}(T) \neq \emptyset$. Let $0 \lt \mu \lt 2 \eta/\kappa^2$ and $0 \leq \gamma L \lt \tau$, where $\tau = 1 - \sqrt{1-\mu(2\eta-\mu\kappa^2)}$. Consider the hierarchical monotone variational inequality problem (in short, HMVIP):

VI (a): finding $z^* \in {\rm Fix}(T)$ such that $\langle(I-V)z^*, z-z^*\rangle \geq 0$, $\forall z \in {\rm Fix}(T)$;

VI (b): finding $x^* \in S$ such that $\langle(\mu F - \gamma f) x^*, x-x^*\rangle \geq 0$, $\forall z \in S$.

Here $S$ denotes the nonempty solution set of the VI (a). This paper combines hybrid steepest-descent method, viscosity method and projection method to design an explicit algorithm, that can be used to find the unique solution of the HMVIP. Strong convergence of the algorithm is proved under very mild conditions. Applications in hierarchical minimization problems are also included.