In this paper, we propose a novel restart scheme for FISTA (Fast Iterative Shrinking-Threshold Algorithm). This method which is a generalization of Nesterov's accelerated gradient algorithm is widely used in the field of large convex optimization problems and it provides fast convergence results under a strong convexity assumption. These convergence rates can be extended for weaker hypotheses such as the \L{}ojasiewicz property but it requires prior knowledge on the function of interest. In particular, most of the schemes providing a fast convergence for non-strongly convex functions satisfying a quadratic growth condition involve the growth parameter which is generally not known. Recent works by Alamo et al. show that restarting FISTA could ensure a fast convergence for this class of functions without requiring any geometry parameter. We improve these restart schemes by providing a better asymptotical convergence rate and by requiring a lower computation cost. We present numerical results emphasizing that our method is efficient especially in terms of computation time
