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Fast Krasnosel'skii-Mann algorithm with a convergence rate of the fixed point iteration of $o\left(\frac{1}{k}\right)$
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abstract
The Krasnosel'skii-Mann (KM) algorithm is the most fundamental iterative scheme designed to find a fixed point of an averaged operator in the framework of a real Hilbert space, since it lies at the heart of various numerical algorithms for solving monotone inclusions and convex optimization problems. We enhance the Krasnosel'skii-Mann algorithm with Nesterov's momentum updates and show that the resulting numerical method exhibits a convergence rate for the fixed point residual of $o(1/k)$ while preserving the weak convergence of the iterates to a fixed point of the operator. Numerical experiments illustrate the superiority of the resulting so-called Fast KM algorithm over various fixed point iterative schemes, and also its oscillatory behavior, which is a specific of Nesterov's momentum optimization algorithms.
Forward citations
Cited by 3 Pith papers
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Variance-Reduced Fast Operator Splitting Methods for Generalized Equations
A new class of accelerated forward-backward and backward-forward splitting methods with variance-reduced estimators achieves O(1/k^2) and o(1/k^2) expected residual convergence for generalized equations under co-coerc...
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Accelerated Extragradient-Type Methods -- Part 2: Generalization and Sublinear Convergence Rates under Co-Hypomonotonicity
A unified family of anchored and Nesterov-accelerated extragradient methods achieves O(1/k) and o(1/k) last-iterate residual convergence for co-hypomonotone inclusions, with iterate convergence for the new GAEG+ class.
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On the Rate of Asymptotic Regularity of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization
KM and Halpern-type iterations in CAT(0) spaces have the same O(1/√n) and O(1/n) asymptotic-regularity rates as in Banach spaces, and a proximal Halpern scheme is proposed for hyperbolic optimization.
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