  We introduce new optimized first-order methods for smooth unconstrained
convex minimization. Drori and Teboulle recently described a numerical method
for computing the $N$-iteration optimal step coefficients in a class of
first-order algorithms that includes gradient methods, heavy-ball methods, and
Nesterov's fast gradient methods. However, Drori and Teboulle's numerical
method is computationally expensive for large $N$, and the corresponding
numerically optimized first-order algorithm requires impractical memory and
computation for large-scale optimization problems. In this paper, we propose
optimized first-order algorithms that achieve a convergence bound that is two
times smaller than for Nesterov's fast gradient methods; our bound is found
analytically and refines the numerical bound. Furthermore, the proposed
optimized first-order methods have efficient recursive forms that are
remarkably similar to Nesterov's fast gradient methods.
