# Polyak Minorant Method for Convex Optimization

arXiv: https://arxiv.org/abs/2310.07922 · HTML: https://mlsanity.com/paper/2310.07922/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2023.

## Built on by (most cited first, 2 of 2)

- [Sparse Polyak: an adaptive step size rule for high-dimensional M-estimation](https://mlsanity.com/paper/2509.09802/) (2025)
- [Local LMO is Secretly a Projection Method!](https://mlsanity.com/paper/2609.33383/) (2026)

## Abstract (arXiv)

In 1963 Boris Polyak suggested a particular step size for gradient descent methods, now known as the Polyak step size, that he later adapted to subgradient methods. The Polyak step size requires knowledge of the optimal value of the minimization problem, which is a strong assumption but one that holds for several important problems. In this paper we extend Polyak's method to handle constraints and, as a generalization of subgradients, general minorants, which are convex functions that tightly lower bound the objective and constraint functions. We refer to this algorithm as the Polyak Minorant Method (PMM). It is closely related to cutting-plane and bundle methods.