Abstract:In this paper, we propose a Two-step Krasnosel'skii-Mann (KM) Algorithm (TKMA) with adaptive momentum for solving convex optimization problems arising in image processing. Such optimization problems can often be reformulated as fixed-point problems for certain operators, which are then solved using iterative methods based on the same operator, including the KM iteration, to ultimately obtain the solution to the original optimization problem. Prior to developing TKMA, we first introduce a KM iteration enhanced with adaptive momentum, derived from geometric properties of an $\alpha$-averaged nonexpansive operator T with $\alpha\in(0,1)$, KM acceleration technique, and information from the composite operator $T^2$. The proposed TKMA is constructed as a convex combination of this adaptive-momentum KM iteration and the Picard iteration of $T^2$. We prove that the sequence generated by TKMA converges weakly to a fixed point of T in a real Hilbert space. Moreover, under $\alpha\in(0,1/2]$ and specific assumptions on the adaptive momentum parameters, we prove that the algorithm achieves an $o\left(1/\sqrt{k}\right)$ convergence rate in terms of the distance between successive iterates. Numerical experiments demonstrate that TKMA outperforms the FPPA, PGA, Fast KM algorithm, and Halpern algorithm on tasks such as image denoising and low-rank matrix completion.
From: Yizun Lin [view email]
[v1]
Tue, 28 Oct 2025 15:44:48 UTC (1,528 KB)
[v2]
Thu, 28 May 2026 13:22:09 UTC (3,749 KB)
[v3]
Tue, 14 Jul 2026 14:27:51 UTC (3,753 KB)
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