# Tangencies and Polynomial Optimization

arXiv: https://arxiv.org/abs/1902.06041 · HTML: https://mlsanity.com/paper/1902.06041/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Feb 2019.

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

- [Iteratively Refined Image Reconstruction with Learned Attentive Regularizers](https://mlsanity.com/paper/2407.06608/) (2024)
- [Creating walls to avoid unwanted points in root finding and optimization](https://mlsanity.com/paper/2309.11475/) (2023)

## Abstract (arXiv)

Given a polynomial function f colon ℝn rightarrow ℝ and a unbounded basic closed semi-algebraic set S subset ℝn, in this paper we show that the conditions listed below are characterized exactly in terms of the so-called em tangency variety of f on S: (i) The f is bounded from below on S; (ii) The f attains its infimum on S; (iii) The sublevel set x ∈ S  |  f(x) ≤ λ for λ ∈ ℝ is compact; (iv) The f is coercive on S. Besides, we also provide some stability criteria for boundedness and coercivity of f on S.