Fej\'er* monotonicity in optimization algorithms

TL;DR

This paper introduces and explores Fejér* monotonicity, an extension of Fejér monotonicity, analyzing its convergence properties and differences from related concepts in optimization algorithms within Hilbert spaces.

Contribution

The paper extends the concept of Fejér monotonicity to Fejér* monotonicity, providing analysis of its behavior, convergence properties, and distinctions from quasi-Fejér notions.

Findings

Fejér* monotonicity exhibits specific convergence behaviors in Hilbert spaces.

Illustrative examples demonstrate differences between Fejér* and quasi-Fejér monotonicity.

The study provides insights into weak and strong convergence of Fejér* sequences.

Abstract

Fej\'er monotonicity is a well-established property often observed in sequences generated by optimization algorithms. In this paper, we study an extension of this property, called Fej\'er* monotonicity, which was initially proposed in [SIAM J. Optim., 34(3), 2535-2556 (2024)]. We discuss and explore its behavior within Hilbert spaces as a tool for optimization algorithms. Additionally, we investigate weak and strong convergence properties of this novel concept. Through illustrative examples and insightful results, we contrast Fej\'er* with weaker notions of quasi-Fej\'er-type monotonicity.