  This paper presents a new convergent Plug-and-Play (PnP) algorithm. PnP
methods are efficient iterative algorithms for solving image inverse problems
formulated as the minimization of the sum of a data-fidelity term and a
regularization term. PnP methods perform regularization by plugging a
pre-trained denoiser in a proximal algorithm, such as Proximal Gradient Descent
(PGD). To ensure convergence of PnP schemes, many works study specific
parametrizations of deep denoisers. However, existing results require either
unverifiable or suboptimal hypotheses on the denoiser, or assume restrictive
conditions on the parameters of the inverse problem. Observing that these
limitations can be due to the proximal algorithm in use, we study a relaxed
version of the PGD algorithm for minimizing the sum of a convex function and a
weakly convex one. When plugged with a relaxed proximal denoiser, we show that
the proposed PnP-$\alpha$PGD algorithm converges for a wider range of
regularization parameters, thus allowing more accurate image restoration.
