  This paper generalizes the optimized gradient method (OGM) that achieves the
optimal worst-case cost function bound of first-order methods for smooth convex
minimization. Specifically, this paper studies a generalized formulation of OGM
and analyzes its worst-case rates in terms of both the function value and the
norm of the function gradient. This paper also develops a new algorithm called
OGM-OG that is in the generalized family of OGM and that has the best known
analytical worst-case bound with rate $O(1/N^{1.5})$ on the decrease of the
gradient norm among fixed-step first-order methods. This paper also proves that
Nesterov's fast gradient method has an $O(1/N^{1.5})$ worst-case gradient norm
rate but with constant larger than OGM-OG. The proof is based on the worst-case
analysis called Performance Estimation Problem.
