  The first part of this work established the foundations of a radial duality
between nonnegative optimization problems, inspired by the work of Renegar
\cite{Renegar2016}. Here we utilize our radial duality theory to design and
analyze projection-free optimization algorithms that operate by solving a
radially dual problem. In particular, we consider radial subgradient,
smoothing, and accelerated methods that are capable of solving a range of
constrained convex and nonconvex optimization problems and that can scale-up
more efficiently than their classic counterparts. These algorithms enjoy the
same benefits as their predecessors, avoiding Lipschitz continuity assumptions
and costly orthogonal projections, in our newfound, broader context. Our radial
duality further allows us to understand the effects and benefits of smoothness
and growth conditions on the radial dual and consequently on our radial
algorithms.
