# Positivity certificates and polynomial optimization on non-compact semialgebraic sets

arXiv: https://arxiv.org/abs/1911.11428 · HTML: https://mlsanity.com/paper/1911.11428/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Nov 2019.

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

- [Tractable hierarchies of convex relaxations for polynomial optimization on the nonnegative orthant](https://mlsanity.com/paper/2209.06175/) (2022)

## Abstract (arXiv)

In a first contribution, we revisit two certificates of positivity on (possibly non-compact) basic semialgebraic sets due to Putinar and Vasilescu [Comptes Rendus de l'Acad'emie des Sciences-Series I-Mathematics, 328(6) (1999) pp. 495-499]. We use Jacobi's technique from [Mathematische Zeitschrift, 237(2) (2001) pp. 259-273] to provide an alternative proof with an effective degree bound on the sums of squares multipliers in such certificates. As a consequence, it allows one to define a hierarchy of semidefinite relaxations for a general polynomial optimization problem. Convergence of this hierarchy to a neighborhood of the optimal value as well as strong duality and analysis are guaranteed. In a second contribution, we introduce a new numerical method for solving systems of polynomial inequalities and equalities with possibly uncountably many solutions. As a bonus, one may apply this method to obtain approximate global optimizers in polynomial optimization.