Regularity of the Product of Two Relaxed Cutters with Relaxation Parameters Beyond Two
Andrzej Cegielski, Simeon Reich, Rafał Zalas
TL;DR
The paper tackles the common fixed-point problem for two $\lambda$- and $\mu$-relaxed cutters with $\lambda\mu<4$, where one relaxation may exceed 2 and yield demicontractive behavior. It proves that the product $UT$ inherits weak and linear regularity properties from $T$ and $U$ on suitable balls, with an explicit modulus formula that depends on $\lambda,\mu$ and the regularity constants; this enables convergence analysis for the fixed-point iteration $x^{k+1}=x^k+\frac{\alpha_k}{\nu}(UT(x^k)-x^k)$ and its variants. The main results extend known preservation of regularity to the broader regime $\lambda\mu<4$, including demicontractive products, and are complemented by corollaries for projection-based problems and generalized Douglas–Rachford, plus a numerical example validating the theory. Overall, the work provides a rigorous framework for designing and analyzing iterative schemes using products of relaxed cutters with parameters beyond the standard two, with quantitative convergence guarantees and practical guidance for feasibility problems.
Abstract
We study the product of two relaxed cutters having a common fixed point. We assume that one of the relaxation parameters is greater than two so that the corresponding relaxed cutter is no longer quasi-nonexpansive, but rather demicontractive. We show that if both of the operators are (weakly/linearly) regular, then under certain conditions, the resulting product inherits the same type of regularity. We then apply these results to proving convergence in the weak, norm and linear sense of algorithms that employ such products.