In this paper, the aim is to design a new proximal gradient algorithm by using the inertial technique with adaptive stepsize for solving convex minimization problems and prove convergence of the iterates under some suitable assumptions. Some numerical implementations of image deblurring are performed to show the efficiency of the proposed methods.

In this study, we suggest a new class of forward-backward (FB) algorithms designed to solve convex minimization problems. Our method incorporates a linesearch technique, eliminating the need to choose Lipschitz assumptions explicitly. Additionally, we apply double inertial extrapolations to enhance the algorithm's convergence rate. We establish a weak convergence theorem under some mild conditions. Furthermore, we perform numerical tests, and apply the algorithm to image restoration and data classification as a practical application. The experimental results show our approach's superior performance and effectiveness, surpassing some existing methods in the literature.