Radial Duality Part II: Applications and Algorithms

TL;DR

This paper develops and applies a radial duality framework to create projection-free optimization algorithms that efficiently solve constrained convex and nonconvex problems without relying on Lipschitz continuity or orthogonal projections.

Contribution

It introduces a novel radial duality theory and leverages it to design scalable, projection-free optimization algorithms with broad applicability.

Findings

Radial duality enables understanding of smoothness effects on algorithms.

Proposed algorithms avoid costly projections and Lipschitz assumptions.

Algorithms are effective for both convex and nonconvex constrained problems.

Abstract

The first part of this work established the foundations of a radial duality between nonnegative optimization problems, inspired by the work of (Renegar, 2016). 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.