  (Renegar, 2016) introduced a novel approach to transforming generic conic
optimization problems into unconstrained, uniformly Lipschitz continuous
minimization. We introduce radial transformations generalizing these ideas,
equipped with an entirely new motivation and development that avoids any
reliance on convex cones or functions. Perhaps of greatest 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 rise to a powerful 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 strong foundation for designing projection-free
radial optimization algorithms, which is carried out in the second part of this
work.
