[Paper Review] A note on an alternative iterative method for nonexpansive mappings

Laurenţiu Leuştean, Adriana Nicolae|arXiv (Cornell University)|Apr 7, 2015

Optimization and Variational Analysis11 references3 citations

TL;DR

This paper demonstrates that an alternative iterative method for nonexpansive mappings in geodesic spaces is fundamentally equivalent to the well-known Halpern iteration. By showing that the sequence generated by the alternative method is a reparameterization of the Halpern iteration, the authors establish that all strong convergence and asymptotic regularity results for Halpern's method directly apply to the alternative method without modification.

ABSTRACT

In this note we point out that results on the asymptotic behaviour of an alternative iterative method are corollaries of corresponding results on the well-known Halpern iteration.

Motivation & Objective

• To clarify the relationship between an alternative iterative method for nonexpansive mappings and the classical Halpern iteration.
• To resolve ambiguity in the literature regarding the convergence behavior of the alternative iterative scheme.
• To establish that results on strong convergence and asymptotic regularity for Halpern's iteration apply directly to the alternative method.
• To provide a theoretical foundation for reusing existing quantitative convergence results in the context of the alternative iterative method.

Proposed method

• Define the alternative iterative sequence $ x_{n+1} = T( u_{n+1}u + (1- u_{n+1})x_n) $, where $ T $ is nonexpansive and $ (\nu_n) \subseteq [0,1] $.
• Introduce an auxiliary sequence $ y_{n+1} = \nu_{n+1}u + (1-\nu_{n+1})x_n $, which satisfies $ x_n = T y_n $ for $ n \geq 1 $.
• Prove that $ (y_n) $ satisfies the Halpern iteration recurrence: $ y_{n+1} = \nu_{n+1}u + (1-\nu_{n+1})T y_n $.
• Use the nonexpansiveness of $ T $ to derive the inequality $ d(x_n, p) \leq d(y_n, p) $ for any fixed point $ p $ of $ T $.
• Establish that $ d(x_m, x_n) \leq d(y_m, y_n) $, implying that convergence and regularity properties of $ (y_n) $ transfer to $ (x_n) $.
• Leverage known results on Halpern iteration to conclude that strong convergence and asymptotic regularity of $ (y_n) $ imply the same for $ (x_n) $.

Experimental results

Research questions

• RQ1 Is the alternative iterative method for nonexpansive mappings in geodesic spaces fundamentally different from the Halpern iteration?
• RQ2 Can convergence and regularity results for the Halpern iteration be transferred to the alternative iterative method?
• RQ3 Does the alternative method inherit the same rate of asymptotic regularity as the Halpern iteration?
• RQ4 Under what conditions does the alternative iterative sequence converge strongly to a fixed point of $ T $?
• RQ5 Can quantitative convergence results for Halpern's iteration be applied without modification to the alternative method?

Key findings

• The sequence $ (y_n) $ defined by $ y_{n+1} = \nu_{n+1}u + (1-\nu_{n+1})x_n $ is the standard Halpern iteration.
• The sequence $ (x_n) $ generated by the alternative method satisfies $ x_n = T y_n $ for all $ n \geq 1 $, making it a reparameterization of the Halpern sequence.
• If $ (y_n) $ converges strongly to a fixed point $ p $ of $ T $, then $ (x_n) $ also converges strongly to $ p $.
• The asymptotic regularity of $ (y_n) $ with rate $ \Phi $ implies the same rate of asymptotic regularity for $ (x_n) $, preserving quantitative bounds.
• All strong convergence and asymptotic regularity results for the Halpern iteration are directly applicable to the alternative iterative method.
• The main result of [13, Theorem 4.1] on the alternative method is a corollary of known results on the Halpern iteration, including [12, Theorem 3.1] and [10, 9].