# Radial Subgradient Method

arXiv: https://arxiv.org/abs/1703.09280 · HTML: https://mlsanity.com/paper/1703.09280/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Mar 2017.

## Built on by (most cited first, 6 of 6)

- [Gauges and Accelerated Optimization over Smooth and/or Strongly Convex Sets](https://mlsanity.com/paper/2303.05037/) (2023)
- [Revisiting Subgradient Method: Complexity and Convergence Beyond Lipschitz Continuity](https://mlsanity.com/paper/2305.14161/) (2023)
- [A Unified Analysis on the Subgradient Upper Bounds for the Subgradient Methods Minimizing Composite Nonconvex, Nonsmooth and Non-Lipschitz Functions](https://mlsanity.com/paper/2308.16362/) (2023)
- [Randomized Coordinate Subgradient Method for Nonsmooth Composite Optimization](https://mlsanity.com/paper/2206.14981/) (2022)
- [Optimization on a Finer Scale: Bounded Local Subgradient Variation Perspective](https://mlsanity.com/paper/2403.16317/) (2024)
- [Some Primal-Dual Theory for Subgradient Methods for Strongly Convex Optimization](https://mlsanity.com/paper/2305.17323/) (2023)

## Abstract (arXiv)

We present a subgradient method for minimizing non-smooth, non-Lipschitz convex optimization problems. The only structure assumed is that a strictly feasible point is known. We extend the work of Renegar [5] by taking a different perspective, leading to an algorithm which is conceptually more natural, has notably improved convergence rates, and for which the analysis is surprisingly simple. At each iteration, the algorithm takes a subgradient step and then performs a line search to move radially towards (or away from) the known feasible point. Our convergence results have striking similarities to those of traditional methods that require Lipschitz continuity. Costly orthogonal projections typical of subgradient methods are entirely avoided.