Hybrid methods for a class of monotone variational inequalities

Abstract

We use contractions to regularize a class of monotone variational inequalities, where the monotone operators are complements of nonexpansive mappings and the solutions are sought in the set of fixed points of another nonexpansive mapping. Such variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Both implicit and explicit schemes are shown to be strongly convergent. An application in hierarchical minimization is included.

Introduction

Let $H$ be a Hilbert space, $C$ a nonempty closed convex subset of $H$, and $F:C\to H$ a nonlinear mapping. A variational inequality problem, denoted $VI(F,C)$, is to find a point $x^{\ast }$ with the property

$$x^{\ast }\in C\text{such that}〈F{x}^{\ast },x-x^{\ast }〉\ge 0\text{,}x\in C\text{.}$$

We say that $VI(F,C)$ is monotone if the mapping $F$ is a monotone operator. In this paper we are concerned with a special class of variational inequalities in which the mapping $F$ is the complement of a nonexpansive mapping and the constraint set is the set of fixed points of another nonexpansive mapping. Namely, we consider the following type of monotone variational inequality problem

$$\text{Find }{x}^{\ast }\in Fix\left(T\right)\text{ such that }〈(I-V)x^{\ast },x-x^{\ast }〉\ge 0\text{,}x\in Fix\left(T\right)\text{,}$$

where $T,V:C\to C$ are nonexpansive mappings. It is always assumed that the set of fixed points of $T$, $Fix\left(T\right)=\{x\in C:Tx=x\}$, is not empty.

It is well-known that the VIP (1.1) is equivalent to the fixed point equation

$$x^{\ast }=P_C(I-\gamma F)x^{\ast }$$

where $\gamma >0$ and $P_C$ is the metric projection of $H$ onto $C$.

It is also well-known that if $F$ is Lipschitzian and strongly monotone, then for small enough $\gamma >0$, the mapping $P_C(I-\gamma F)$ is a contraction on $C$ and so the sequence $\left\{x_n\right\}$ of Picard iterates, given by $x_n=P_C(I-\gamma F)x_{n-1}$ ($n\ge 1$), converges strongly to the unique solution of the VIP (1.1).

Hybrid methods for solving the variational inequality (1.1) were studied by Yamada [1] where he assumed that $F$ is Lipschitzian and strongly monotone. However, his methods do not apply to the variational inequality (1.2) since the mapping $I-V$ fails, in general, to be strongly monotone, though it is Lipschitzian. Therefore, other hybrid methods have to be sought.

Recently, Mainge and Moudafi [2] introduced a hybrid iterative method for solving the variational inequality (1.2), which generates a sequence $\left(x_n\right)$ as follows:

$$x_{n+1}={\lambda }_{n}f\left(x_n\right)+(1-{\lambda }_n)[{\alpha }_{n}V{x}_n+(1-{\lambda }_n)T{x}_n]\text{,}$$

where the initial guess $x_0\in C$, $f:C\to C$ is a contraction, and $\left({\lambda }_n\right)$ and $\left({\alpha }_n\right)$ are sequences in $[0,1]$ satisfying certain conditions.

It is the purpose of this paper to further investigate other hybrid iterative methods for solving the variational inequality (1.2). More precisely, assuming (1.2) is consistent and noticing the fact that if, for each $t\in (0,1)$, $x_t\in C$ is a fixed point of the nonexpansive mapping

$$tV+(1-t)T\text{,}$$

then every weak accumulation point of $\left(x_t\right)$ as $t\to 0$ is a solution of (1.2), we are able, upon the idea of regularization, to introduce a new hybrid iterative method as follows:

$$z_{n+1}={\lambda }_n[{\alpha }_{n}f\left(z_n\right)+(1-{\alpha }_n)V{z}_n]+(1-{\lambda }_n)T{z}_n\text{,}$$

where $\left\{{\lambda }_n\right\}$ and $\left\{{\alpha }_n\right\}$ are sequences in (0, 1), and $f:C\to C$ is a contraction. Our idea is to regularize the nonexpansive mapping $V$, instead of the nonexpansive mapping $W_t≔tV+(1-t)T$ as done by Moudafi and Mainge [3]. Since Moudafi and Mainge’s regularization depends upon $t$ whereas ours not, we obtain our convergence result for the regularization under dramatically less restrictive conditions; as a matter of fact, the conditions (A1) and (A3) of Moudafi and Mainge [3] are completely removed.

We also apply both of our implicit and explicit schemes to solve a hierarchical minimization problem in a Hilbert space.