  (Renegar, 2016) introduced a novel approach to transforming generic conic
optimization problems into unconstrained, uniformly Lipschitz continuous
minimization. We introduce {\it radial transformations} generalizing these
ideas, equipped with an entirely new motivation and development that avoids any
reliance on convex cones or functions. Of practical importance, this
facilitates the development of new families of projection-free first-order
methods applicable even in the presence of nonconvex objectives and constraint
sets. Our generalized construction of this radial transformation uncovers that
it is dual (i.e., self-inverse) for a wide range of functions including all
concave objectives. This gives a new duality relating optimization problems to
their radially dual problem. For a broad class of functions, we characterize
continuity, differentiability, and convexity under the radial transformation as
well as develop a calculus for it. This radial duality provides a foundation
for designing projection-free radial optimization algorithms, which is carried
out in the second part of this work.
