On the String Averaging Method for Sparse Common Fixed Points Problems.

Area of Science:

• Optimization Theory
• Nonlinear Analysis
• Applied Mathematics

Background:

• The common fixed point problem is central to many areas of applied mathematics and optimization.
• Directed operators, a significant class of nonlinear operators, are frequently encountered in convex optimization algorithms.
• Existing methods may face challenges with the efficiency and convergence for specific operator families.

Purpose of the Study:

• To define and analyze the concept of sparseness for families of directed operators.
• To develop and investigate a novel algorithmic scheme for solving common fixed-point problems involving sparse operator families.
• To adapt and extend these methods to address the convex feasibility problem.

Main Methods:

• Introduced a formal definition of sparseness for families of directed operators.
• Developed and analyzed a string-averaging algorithmic scheme tailored for sparse operator families.
• Applied the framework to the convex feasibility problem, deriving a new subgradient projection algorithm.

Main Results:

• Demonstrated the effectiveness of the string-averaging algorithm for solving common fixed-point problems with sparse directed operators.
• Established favorable convergence properties for the proposed algorithmic scheme.
• Obtained a new subgradient projection algorithm for the convex feasibility problem as a special case.

Conclusions:

• The proposed definition of sparseness provides a valuable characterization for operator families in optimization.
• The string-averaging algorithm offers an efficient approach for tackling common fixed-point problems with sparse operators.
• The study extends the applicability of these concepts to the important convex feasibility problem.