# "Efficient" Subgradient Methods for General Convex Optimization

arXiv: https://arxiv.org/abs/1605.08712 · HTML: https://mlsanity.com/paper/1605.08712/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2016.

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

- [Restarts subject to approximate sharpness: A parameter-free and optimal scheme for first-order methods](https://mlsanity.com/paper/2301.02268/) (2023)
- [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)

## Abstract (arXiv)

A subgradient method is presented for solving general convex optimization problems, the main requirement being that a strictly-feasible point is known. A feasible sequence of iterates is generated, which converges to within user-specified error of optimality. Feasibility is maintained with a line-search at each iteration, avoiding the need for orthogonal projections onto the feasible region (the operation that limits practicality of traditional subgradient methods). Lipschitz continuity is not required, yet the algorithm is shown to possess a convergence rate analogous to rates for traditional methods, albeit with error measured relatively, whereas traditionally error has been absolute. The algorithm is derived using an elementary framework that can be utilized to design other such algorithms.