# Radial Duality Part II: Applications and Algorithms

arXiv: https://arxiv.org/abs/2104.11185 · HTML: https://mlsanity.com/paper/2104.11185/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Apr 2021.

## Built on by (most cited first, 4 of 4)

- [Gauges and Accelerated Optimization over Smooth and/or Strongly Convex Sets](https://mlsanity.com/paper/2303.05037/) (2023)
- [Some Primal-Dual Theory for Subgradient Methods for Strongly Convex Optimization](https://mlsanity.com/paper/2305.17323/) (2023)
- [Nonsmooth Projection-Free Optimization with Functional Constraints](https://mlsanity.com/paper/2311.11180/) (2023)
- [An Elementary Proof of the Near Optimality of LogSumExp Smoothing](https://mlsanity.com/paper/2512.10825/) (2025)

## Abstract (arXiv)

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.