Sparse string-averaging and split common fixed points.

*(English)* Zbl 1229.47107
Leizarowitz, Arie (ed.) et al., Nonlinear analysis and optimization I. Nonlinear analysis. A conference in celebration of Alex Ioffe’s 70th and Simeon Reich’s 60th birthdays, Haifa, Israel, June 18–24, 2008. Providence, RI: American Mathematical Society (AMS); Ramat-Gan: Bar-Ilan University (ISBN 978-0-8218-4834-0/pbk). Contemporary Mathematics 513, 125-142 (2010).

Summary: We review the common fixed point problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We present our recent definition of sparseness of a family of operators and discuss a string-averaging algorithmic scheme that favorably handles the common fixed point problem when the family of operators is sparse. We also review some recent results on the multiple operators split common fixed point problem which requires to find a common fixed point of a family of operators in one space whose image under a linear transformation is a common fixed point of another family of operators in the image space.

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47N10 | Applications of operator theory in optimization, convex analysis, programming, economics |

90C25 | Convex programming |