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String-averaging is an algorithmic structure used when handling a family of operators in situations where the algorithm in hand requires to employ the operators in a specific order. Sequential orderings are well known, and a simultaneous order means that all operators are used simultaneously (in parallel). String-averaging allows to use strings of indices, constructed by subsets of the index set of all operators, to apply the operators along these strings, and then to combine their end-points in some agreed manner to yield the next iterate of the algorithm. String-averaging methods were discussed and used for solving the common fixed point problem or its important special case of the convex feasibility problem. In this paper we propose and investigate string-averaging methods for the problem of best approximation to the common fixed point set of a family of operators. This problem involves finding a point in the common fixed point set of a family of operators that is closest to a given point, called an anchor point, in contrast with the common fixed point problem that seeks any point in the common fixed point set. We construct string-averaging methods for solving the best approximation problem to the common fixed points set of either finite or infinite families of firmly nonexpansive operators in a real Hilbert space. We show that the simultaneous Halpern–Lions–Wittman–Bauschke algorithm, the Halpern–Wittman algorithm, and the Combettes algorithm, which were not labeled as string-averaging methods, are actually special cases of these methods. Some of our string-averaging methods are labeled as “static” because they use a fixed pre-determined set of strings. Others are labeled as “quasi-dynamic” because they allow the choices of strings to vary, between iterations, in a specific manner and belong to a finite fixed pre-determined set of applicable strings. For the problem of best approximation to the common fixed point set of a family of operators, the full dynamic case that would allow strings to unconditionally vary between iterations remains unsolved, although it exists and is validated in the literature for the convex feasibility problem where it is called “dynamic string-averaging”.

String-averaging methods for best approximation to common fixed point sets of operators: the finite and infinite cases