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”.

中文翻译：

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字符串平均方法，用于最佳逼近运算符的公共不动点集：有限和无穷大情况

字符串平均是在现有算法要求以特定顺序使用运算符的情况下处理一系列运算符时使用的算法结构。顺序排序是众所周知的，并且同时排序意味着同时（并行）使用所有运算符。字符串平均允许使用由所有运算符的索引集的子集构成的索引字符串，将运算符沿这些字符串应用，然后以某种协商的方式组合其端点，以产生算法的下一个迭代。讨论了字符串平均方法，并将其用于解决公共不动点问题或其凸可行性问题的重要特例。在本文中，我们提出并研究了字符串平均方法，以解决对算子族的公共不动点集的最佳逼近问题。与在公共定点集中寻找任意点的公共定点问题相反，此问题涉及在一组算子的公共定点集中找到最接近给定点的点，称为锚点。我们构造了字符串平均方法，用于解决实际希尔伯特空间中有限非固定算子的有限或无限族的公共不动点集的最佳逼近问题。我们证明没有被标记为字符串平均方法的并发的Halpern-Lions-Wittman-Bauschke算法，Halpern-Wittman算法和Combettes算法实际上是这些方法的特例。我们的某些字符串平均方法被标记为“静态”，因为它们使用一组固定的预定字符串。其他一些被标记为“准动态”，因为它们允许字符串的选择以特定的方式在迭代之间变化，并且属于有限固定的预定适用字符串集合。对于最佳近似于一个算子族的公共不动点集的问题，允许字符串在迭代之间无条件变化的全动态情况仍未解决，尽管它存在并在文献中针对凸可行性问题进行了验证。称为“动态字符串平均”。在迭代之间，以特定的方式，并且属于有限固定的，适用的字符串的预定集合。对于最佳近似于一个算子族的公共不动点集的问题，允许字符串在迭代之间无条件变化的全动态情况仍未解决，尽管它存在并在文献中针对凸可行性问题进行了验证。称为“动态字符串平均”。在迭代之间，以特定的方式，并且属于有限固定的，适用的字符串的预定集合。对于最佳近似于一个算子族的公共不动点集的问题，允许字符串在迭代之间无条件变化的全动态情况仍未解决，尽管它存在并在文献中针对凸可行性问题进行了验证。称为“动态字符串平均”。