# The Hierarchy of Local Minimums in Polynomial Optimization

arXiv: https://arxiv.org/abs/1311.4228 · HTML: https://mlsanity.com/paper/1311.4228/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2013.

## Built on by (most cited first, 1 of 1)

- [Escaping Saddle Points Faster with Stochastic Momentum](https://mlsanity.com/paper/2106.02985/) (2021)

## Abstract (arXiv)

This paper studies the hierarchy of local minimums of a polynomial in the space. For this purpose, we first compute H-minimums, for which the first and second order optimality conditions are satisfied. To compute each H-minimum, we construct a sequence of semidefinite relaxations, based on optimality conditions. We prove that each constructed sequence has finite convergence, under some generic conditions. A procedure for computing all local minimums is given. When there are equality constraints, we have similar results for computing the hierarchy of critical values and the hierarchy of local minimums. Several extensions are discussed.