Methods for Variational Inequality Problem Over the Intersection of Fixed Point Sets of Quasi-Nonexpansive Operators

Abstract

Many convex optimization problems in a Hilbert space ℋ can be written as the following variational inequality problem VIP (ℱ, C): Find such that for all z ∈ C, where C ⊂ ℋ is closed convex and ℱ: ℋ → ℋ is monotone. We consider a special case of VIP (ℱ, C), where and U _i : ℋ → ℋ are quasi-nonexpansive operators having a common fixed point, i ∈ I: = {1, 2,…, m}. A standard method for VIP (ℱ, C) is the projected gradient method u ^k+1 = P _C (u ^k  − μℱu ^k ) which generates sequences converging to a unique solution of VIP (ℱ, C) if ℱ is strongly monotone and Lipschitz continuous. Unfortunately, the method cannot be applied for , because, in general, P _C u cannot be computed explicitly, u ∈ ℋ. Lions in 1977 and Bauschke in 1996 considered a special case of , where ℱ = Id −a, for some a ∈ ℋ, U _i are firmly nonexpansive or nonexpansive, respectively, and studied the convergence properties of the following method: u ^k+1 = U _{i k} u ^k  − λ _k ℱU _{i k} u ^k , where λ _k  ↓ 0 and is a cyclic control, i.e., i _k  = k(mod m) +1 for all k ≥ 0 (see [1, 22]). We apply this method in case ℱ is strongly monotone and Lipschitz continuous, U _i are quasi-nonexpansive and is almost-cyclic. We present the method in a more general form

where T _k : ℋ → ℋ, k ≥ 0, are quasi-nonexpansive, and Fix T _k approximate Fix T in some sense. A special case of the method with T _k  = T for all k ≥ 0 was studied in Yamada and Ogura [32] and by Yamada in [31], (in the latter paper, T was supposed to be nonexpansive). We give sufficient conditions for the convergence of (1) as well as present examples of methods which satisfy these conditions.

Keywords:

• Fixed point
• Quasi-nonexpansive operator
• Variational inequality problem

Mathematics Subject Classification:

• 47H05
• 47H09
• 47H10
• 65K15

ACKNOWLEDGMENTS

The authors thank two anonymous referees for their constructive comments which helped to improve the paper.