  We propose a conditional gradient framework for a composite convex
minimization template with broad applications. Our approach combines smoothing
and homotopy techniques under the CGM framework, and provably achieves the
optimal $\mathcal{O}(1/\sqrt{k})$ convergence rate. We demonstrate that the
same rate holds if the linear subproblems are solved approximately with
additive or multiplicative error. In contrast with the relevant work, we are
able to characterize the convergence when the non-smooth term is an indicator
function. Specific applications of our framework include the non-smooth
minimization, semidefinite programming, and minimization with linear inclusion
constraints over a compact domain. Numerical evidence demonstrates the benefits
of our framework.
