  Adaptive cubic regularization methods have emerged as a credible alternative
to linesearch and trust-region for smooth nonconvex optimization, with optimal
complexity amongst second-order methods. Here we consider a general/new class
of adaptive regularization methods, that use first- or higher-order local
Taylor models of the objective regularized by a(ny) power of the step size and
applied to convexly-constrained optimization problems. We investigate the
worst-case evaluation complexity/global rate of convergence of these
algorithms, when the level of sufficient smoothness of the objective may be
unknown or may even be absent. We find that the methods accurately reflect in
their complexity the degree of smoothness of the objective and satisfy
increasingly better bounds with improving accuracy of the models. The bounds
vary continuously and robustly with respect to the regularization power and
accuracy of the model and the degree of smoothness of the objective.
