  In the development of first-order methods for smooth (resp., composite)
convex optimization problems, where smooth functions with Lipschitz continuous
gradients are minimized, the gradient (resp., gradient mapping) norm becomes a
fundamental optimality measure. Under this measure, a fixed iteration algorithm
with the optimal iteration complexity is known, while determining this number
of iteration to obtain a desired accuracy requires the prior knowledge of the
distance from the initial point to the optimal solution set. In this paper, we
report an adaptive regularization approach, which attains the nearly optimal
iteration complexity without knowing the distance to the optimal solution set.
To obtain further faster convergence adaptively, we secondly apply this
approach to construct a first-order method that is adaptive to the H\"olderian
error bound condition (or equivalently, the {\L}ojasiewicz gradient property),
which covers moderately wide classes of applications. The proposed method
attains nearly optimal iteration complexity with respect to the gradient
mapping norm.
