From Halpern’s fixed-point iterations to Nesterov’s accelerated interpretations for root-finding problems

Quoc Tran-Dinh ()
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Quoc Tran-Dinh: The University of North Carolina at Chapel Hill

Computational Optimization and Applications, 2024, vol. 87, issue 1, No 6, 218 pages

Abstract: Abstract We derive an equivalent form of Halpern’s fixed-point iteration scheme for solving a co-coercive equation (also called a root-finding problem), which can be viewed as a Nesterov’s accelerated interpretation. We show that one method is equivalent to another via a simple transformation, leading to a straightforward convergence proof for Nesterov’s accelerated scheme. Alternatively, we directly establish convergence rates of Nesterov’s accelerated variant, and as a consequence, we obtain a new convergence rate of Halpern’s fixed-point iteration. Next, we apply our results to different methods to solve monotone inclusions, where our convergence guarantees are applied. Since the gradient/forward scheme requires the co-coerciveness of the underlying operator, we derive new Nesterov’s accelerated variants for both recent extra-anchored gradient and past-extra anchored gradient methods in the literature. These variants alleviate the co-coerciveness condition by only assuming the monotonicity and Lipschitz continuity of the underlying operator. Interestingly, our new Nesterov’s accelerated interpretation of the past-extra anchored gradient method involves two past-iterate correction terms. This formulation is expected to guide us developing new Nesterov’s accelerated methods for minimax problems and their continuous views without co-coericiveness. We test our theoretical results on two numerical examples, where the actual convergence rates match well the theoretical ones up to a constant factor.

Keywords: Halpern’s fixed-point iteration; Nesterov’s accelerated method; Co-coercive equation; Maximally monotone inclusion; Extra-anchored gradient method; 90C25; 90C06; 90-08 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10589-023-00518-8

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