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

Many convex optimization problems in a Hilbert space ? can be written as the following variational inequality problemVIP (?,C): Findsuch thatfor allz ? C, whereC ? ? is closed convex and ?: ? ? ? is monotone. We consider a special case ofVIP (?,C), whereandUi: ? ? ? are quasi-nonexpansive operators having a common fixed point,i ? I: = {1, 2,…,m}. A standard method forVIP (?,C) is the projected gradient methoduk+1 = PC(uk ? ??uk) which generates sequences converging to a unique solution ofVIP (?,C) if ? is strongly monotone and Lipschitz continuous. Unfortunately, the method cannot be applied for, because, in general,PCucannot be computed explicitly,u? ?. Lions in 1977 and Bauschke in 1996 considered a special case of, where ? = Id ?a, for somea? ?,Uiare firmly nonexpansive or nonexpansive, respectively, and studied the convergence properties of the following method:uk+1 = Ui kuk ? ?k?Ui kuk, where ?k ? 0 andis a cyclic control, i.e.,ik = k(mod m) +1 for allk ? 0 (see [1, 22]). We apply this method in case ? is strongly monotone and Lipschitz continuous,Uiare quasi-nonexpansive andis almost-cyclic. We present the method in a more general formwhereTk: ? ? ?,k ? 0, are quasi-nonexpansive,and Fix Tkapproximate Fix Tin some sense. A special case of the method withTk = Tfor allk ? 0 was studied in Yamada and Ogura [32] and by Yamada in [31], (in the latter paper,Twas supposed to be nonexpansive). We give sufficient conditions for the convergence of (1) as well as present examples of methods which satisfy these conditions. [ABSTRACT FROM AUTHOR]