  In this paper, acceleration of gradient methods for convex optimization
problems with weak levels of convexity and smoothness is considered. Starting
from the universal fast gradient method which was designed to be an optimal
method for weakly smooth problems whose gradients are H\"older continuous, its
momentum is modified appropriately so that it can also accommodate uniformly
convex and weakly smooth problems. Different from the existing works, fast
gradient methods proposed in this paper do not use the restarting technique but
use momentums that are suitably designed to reflect both the uniform convexity
and the weak smoothness information of the target energy function. Both
theoretical and numerical results that support the superiority of proposed
methods are presented.
