Fej\'er and Fej\'er* Monotonicity: New Results and Limiting Examples

Abstract

Many algorithms in convex optimization and variational analysis can be analyzed using Fej\'er monotone sequences. In 2024, Behling, Bello-Cruz, Iusem, Alves Ribeiro, and Santos introduced a new, more general, notion: Fej\'er* monotonicity. They obtained basic results and discussed applications in optimization. In this work, we complement Behling et al.'s work by presenting a thorough study of Fej\'er* monotonicity. We reveal striking similarities and differences between these notions, including descriptions of the maximal Fej\'er* set. Moreover, we also touch upon Opial sequences and quasi-Fej\'er monotonicity. Throughout this paper, we provide numerous limiting examples and counterexamples.