  The task of finding optimal protocols that minimize the energetic cost of
thermodynamic processes of long yet finite duration $\tau$ is a pressing one.
We approach this problem here in a rigorous and systematic fashion by means of
the adiabatic perturbation theory of closed Hamiltonian quantum systems. Our
main finding is a $1/\tau^2$ scaling of the excess work for large $\tau$ in
gapped systems. This result is at odds with the $1/\tau$ prediction of the
geometric approach to optimization, which is predicated on the slow evolution
of open systems close to canonical equilibrium. In contrast, our approach does
not lead to an obvious geometric interpretation. Furthermore, as the
thermodynamic work does not depend on how an isolated quantum system is split
into a system of interest and its environment, our results imply the failure of
the geometric approach prediction even for open systems. Additionally, we
provide alternative optimization procedures, both for slowly-varying processes
described by adiabatic perturbation theory and for weakly-varying processes
described by linear response theory. Our findings are benchmarked and confirmed
through the application to the driven transverse-field Ising chain.
