  We show that every $\mathbb{R}^d$-valued Sobolev path with regularity
$\alpha$ and integrability $p$ can be lifted to a Sobolev rough path in the
sense of T. Lyons provided $\alpha >1/p>0$. Moreover, we prove the existence of
unique rough path lifts which are optimal w.r.t. strictly convex functionals
among all possible rough path lifts given a Sobolev path. As examples, we
consider the rough path lift with minimal Sobolev norm and characterize the
Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. to a
suitable convex functional. Generalizations of the results to Besov spaces are
briefly discussed.
