Convergence analysis of the Halpern iteration with adaptive anchoring parameters

Abstract

We propose an adaptive way to choose the anchoring parameters for the Halpern iteration to find a fixed point of a nonexpansive mapping in a real Hilbert space. We prove strong convergence of this adaptive Halpern iteration and obtain the rate of asymptotic regularity at least O(1/k), where k is the number of iterations. Numerical experiments are also provided to show advantages and outperformance of our adaptive Halpern algorithm over the standard Halpern algorithm.

Figures (13)

• Figure 1: Increase of $\varphi_k$ with $k$ for Example 5.1.
• Figure 2: Comparison of Algorithm \ref{['Al:3.1']} and Halpern iteration with $\lambda_k=\frac{1}{k+1}$ for Example \ref{['eg:5.1']}.
• Figure 3: Increase of $\varphi_k$ with $k$ for LASSO problem
• Figure 4: Comparison of Algorithm \ref{['Al:3.1']} and Halpern iteration with $\lambda_k=\frac{1}{k+1}$ for Lasso problem with $m=600$, $n=2560$ and $K=100$ for the cameraman.
• Figure 5: Increase of $\varphi_k$ with $k$ for the cameraman.

Theorems & Definitions (22)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Remark 3.1
• Lemma 3.1
• proof
• Theorem 3.1
• proof