Positivity certificates and polynomial optimization on non-compact semialgebraic sets

TL;DR

This paper revisits positivity certificates for non-compact semialgebraic sets, providing effective degree bounds and a hierarchy of semidefinite relaxations for polynomial optimization, along with a new numerical method for solving polynomial systems.

Contribution

It offers an alternative proof with degree bounds for positivity certificates and introduces a novel numerical approach for polynomial systems and optimization.

Findings

Hierarchy of semidefinite relaxations converges to a neighborhood of the optimum.

Strong duality and convergence guarantees are established.

New numerical method effectively solves polynomial inequalities and finds approximate global optimizers.

Abstract

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.