On Opial's Lemma
Aleksandr Arakcheev, Heinz H. Bauschke
TL;DR
The paper investigates convergence properties of Opial sequences in real Hilbert spaces, introducing Opial sequences and Opial sets and situating them alongside Fejér monotone concepts. It derives fundamental results on boundedness and cluster-point structure, provides weak and strong convergence criteria via cluster points in the Opial set, and analyzes the role of affine hull enlargement. The work introduces the asymptotic center $A_C(u_{\mathbb N})$ and studies the behavior of projections $P_C x_n$, showing weak convergence to the asymptotic center under certain conditions and illustrating contrasts with the Fejér-monotone case; it also presents sharp examples to delineate the limits of these results. Overall, the results clarify how Opial sequences converge and how projection dynamics differ from Fejér monotone frameworks, informing algorithmic convergence analysis in optimization.
Abstract
Opial's Lemma is a fundamental result in the convergence analysis of sequences generated by optimization algorithms in real Hilbert spaces. We introduce the concept of Opial sequences - sequences for which the limit of the distance to each point in a given set exists. We systematically derive properties of Opial sequences, contrasting them with the well-studied Fejér monotone sequences, and establish conditions for weak and strong convergence. Key results include characterizations of weak convergence via weak cluster points (reaffirming Opial's Lemma), strong convergence via strong cluster points, and the behavior of projections onto Opial sets in terms of asymptotic centers. Special cases and examples are provided to highlight the subtle differences in convergence behaviour and projection properties compared to the Fejér monotone case.