Extrapolated Plug-and-Play Three-Operator Splitting Methods for Nonconvex Optimization with Applications to Image Restoration

This paper investigates the convergence properties and applications of the three-operator splitting method, also known as the Davis--Yin splitting (DYS) method, integrated with extrapolation and plug-and-play (PnP) denoiser within a nonconvex framework. We first propose an extrapolated DYS method to effectively solve a class of structural nonconvex optimization problems that involve minimizing the sum of three possibly nonconvex functions. Our approach provides an algorithmic framework that encompasses both extrapolated forward-backward splitting and extrapolated Douglas-Rachford splitting methods. To establish the convergence of the proposed method, we rigorously analyze its behavior based on the Kurdyka--\Lojasiewicz property, subject to some tight parameter conditions. Moreover, we introduce two extrapolated PnP-DYS methods with convergence guarantee, where the traditional regularization step is replaced by a gradient step--based denoiser. This denoiser is designed using a differentiable neural network and can be reformulated as the proximal operator of a specific nonconvex functional. We conduct extensive experiments on image deblurring and image superresolution problems, where our numerical results showcase the advantage of the extrapolation strategy and the superior performance of the learning-based model that incorporates the PnP denoiser in terms of achieving high-quality recovery images.