Title: A Generalized Accelerated Composite Gradient Method: Uniting Nesterov's Fast Gradient Method and FISTA,
Abstract: We demonstrate that the augmented estimate sequence framework unites the most
popular primal first-order schemes for large-scale problems: the Fast Gradient
Method (FGM) and the Fast Iterative Shrinkage Thresholding Algorithm (FISTA).
We further showcase the flexibility of the augmented estimate sequence by
deriving a Generalized Accelerated Composite Gradient Method endowed with
monotonicity alongside a versatile line-search procedure. The new method
surpasses both FGM and FISTA in terms of robustness and usability. In
particular, it is guaranteed to converge without requiring any quantitative
prior information on the problem. Additional information, if available, leads
to an improvement in performance at least on par with the state-of-the-art. We
support our findings with simulation results.