  This paper provides a new way of developing the fast iterative
shrinkage/thresholding algorithm (FISTA) that is widely used for minimizing
composite convex functions with a nonsmooth term such as the $\ell_1$
regularizer. In particular, this paper shows that FISTA corresponds to an
optimized approach to accelerating the proximal gradient method with respect to
a worst-case bound of the cost function. This paper then proposes a new
algorithm that is derived by instead optimizing the step coefficients of the
proximal gradient method with respect to a worst-case bound of the composite
gradient mapping. The proof is based on the worst-case analysis called
Performance Estimation Problem.
