A Strongly Convergent Halpern-Inertial Three-Operator Splitting Algorithm with Moving Point Projection and Its Application to Image Inpainting

We propose a Halpern-Inertial Three-Operator Splitting Algorithm with Moving Point Projection (HI-TOS-MPP) for solving monotone inclusion problems involving the sum of two maximally monotone operators and one cocoercive operator in real Hilbert spaces. The framework incorporates adaptive Polyak-type inertial extrapolation into the Davis–Yin splitting scheme, employs a Halpern anchor to ensure strong convergence, and applies a moving point projection (MPP) onto dynamically generated half-spaces enclosing the solution set. Under mild and verifiable conditions on the parameters, we prove the strong convergence of the iterate sequence without imposing compactness assumptions. To assess practical performance, we apply the algorithm to image inpainting under severe micro-scratch degradation, which is formulated via dual-directional matrix and tensor nuclear norm regularizations. Numerical experiments on four benchmark images (two grayscale and two color RGB) damaged by four distinct scratch patterns (straight lines, curved hairline cracks, cross-hatching, and random dashes) show that HI-TOS-MPP consistently outperforms both the classical Davis–Yin three-operator splitting method (Davis–Yin TOS) and the Halpern three-operator splitting method (Halpern-TOS), achieving SNR gains of up to 14.31 dB along with marked improvements in SSIM. Furthermore, comparative benchmarks against the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) on the composite objective demonstrate that our two-operator variant achieves a 2.10× wall-clock speedup to target quality thresholds, while the three-operator HI-TOS-MPP attains the highest overall reconstruction fidelity (22.04 dB). The moderate per-iteration resolvent overhead is justified by the algorithm’s capability to decouple non-separable multi-operator and tensor inclusions where FISTA cannot be applied directly.