Complete characterization of exact optimal fixed-point algorithms

Characterize the complete set of fixed-point algorithms for nonexpansive operators that achieve the exact optimal worst-case rate matching the known lower bound, beyond the family parameterized by H-matrices presented in Section 4, and provide a self-contained description of all such exact optimal methods.

Background

Section 4 introduces an (N−2)-dimensional continuous family of fixed-point algorithms, defined via H-matrix representations, that achieve the exact optimal rate matching lower bounds, and shows that Optimal Halpern Method (OHM) and its dual are boundary points of this family.

The authors note that this family is not exhaustive and that other exact optimal algorithms exist, providing examples in the appendix. They explicitly state that fully characterizing the entire set of exact optimal algorithms is challenging and deferred to future work.