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

TL;DR

This paper provides a comprehensive analysis of Fejér* monotonicity, a generalization of Fejér monotonicity, including its properties, differences, and examples, with implications for convex optimization algorithms.

Contribution

It offers a detailed study of Fejér* monotonicity, highlighting its properties, differences from classical Fejér monotonicity, and providing numerous examples and counterexamples.

Findings

Fejér* monotonicity generalizes Fejér monotonicity with distinct properties.

The maximal Fejér* set is characterized and compared to classical sets.

Numerous limiting examples illustrate the concepts and differences.

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.