  In this paper we consider a composite optimization problem that minimizes the
sum of a weakly smooth function and a convex function with either a bounded
domain or a uniformly convex structure. In particular, we first present a
parameter-dependent conditional gradient method for this problem, whose step
sizes require prior knowledge of the parameters associated with the H\"older
continuity of the gradient of the weakly smooth function, and establish its
rate of convergence. Given that these parameters could be unknown or known but
possibly conservative, such a method may suffer from implementation issue or
slow convergence. We therefore propose a parameter-free conditional gradient
method whose step size is determined by using a constructive local quadratic
upper approximation and an adaptive line search scheme, without using any
problem parameter. We show that this method achieves the same rate of
convergence as the parameter-dependent conditional gradient method. Preliminary
experiments are also conducted and illustrate the superior performance of the
parameter-free conditional gradient method over the methods with some other
step size rules.
