Radial Duality Part I: Foundations

TL;DR

This paper introduces radial transformations that generalize existing conic optimization methods, enabling new projection-free algorithms applicable to nonconvex problems, and establishes a duality framework for these transformations.

Contribution

It develops a new class of radial transformations with a duality property, broadening the scope of projection-free optimization methods beyond convex cones.

Findings

Radial transformations are dual and self-inverse for many functions.

Characterization of continuity, differentiability, and convexity under radial transformations.

Foundation for designing new projection-free optimization algorithms.

Abstract

(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.